A000122 -id:A000122 - cp's OEIS frontend

A004009 Expansion of Eisenstein series E_4(q) (alternate convention E_2(q)); theta series of E_8 lattice.

1, 240, 2160, 6720, 17520, 30240, 60480, 82560, 140400, 181680, 272160, 319680, 490560, 527520, 743040, 846720, 1123440, 1179360, 1635120, 1646400, 2207520, 2311680, 2877120, 2920320, 3931200, 3780240, 4747680, 4905600, 6026880

Offset: 0

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
E_8 is also the Barnes-Wall lattice in 8 dimensions.
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan Lambert series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).
The E_8 lattice is integral, unimodular, and even. The 240 shortest nonzero vectors in the lattice have norm squared 2. Of these vectors, 128 are all half-integer, and 112 are all integer. - Michael Somos, Jun 10 2019

			G.f. = 1 + 240*x + 2160*x^2 + 6720*x^3 + 17520*x^4 + 30240*x^5 + 60480*x^6 + ...
G.f. = 1 + 240*q^2 + 2160*q^4 + 6720*q^6 + 17520*q^8 + 30240*q^10 + 60480*q^12 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 123.
W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 53.
R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 53.
N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 111.
S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 (1916), 11-15 (Eq. (25)). Collected Papers of Srinivasa Ramanujan, Chap. 16, Ed. G. H. Hardy et al., Chelsea, NY, 1962.
S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 (1916), 11-15 (Eq. (25)). Ramanujan's Papers, p. 196, Ed. B. J. Venkatachala et al., Prism Books, Bangalore 2000.
Jean-Pierre Serre, "A Course in Arithmetic", Springer, 1978
Joseph H. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves", Springer, 1994
N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from N. J. A. Sloane)
D. Bump, Automorphic Forms and Representations, Cambr. Univ. Press, 1997, p. 29.
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
Heng Huat Chan, Shaun Cooper, and Pee Choon Toh, Ramanujan's Eisenstein series and powers of Dedekind's eta-function, Journal of the London Mathematical Society 75.1 (2007): 225-242. See Q(q).
Henry Cohn and Stephen D. Miller, Some properties of optimal functions for sphere packing in dimensions 8 and 24, arXiv:1603.04759 [math.MG], 2016.
H. S. M. Coxeter, Integral Cayley numbers, Duke Math. J. 13 (1946), 561-578; reprinted in "Twelve Geometric Essays", pp. 20-39.
D. de Laat and F. Vallentin, A Breakthrough in Sphere Packing: The Search for Magic Functions, arXiv preprint arXiv:1607.02111 [math.MG], 2016.
Yang-Hui He and John McKay, Sporadic and Exceptional, arXiv:1505.06742 [math.AG], 2015.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
M. Kaneko and D. Zagier, Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998
Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
Robert V. Moody and Jiri Patera, Fast recursion formula for weight multiplicities, Bulletin of the American Mathematical Society 7.1 (1982): 237-242.
G. Nebe and N. J. A. Sloane, Home page for E_8 lattice
H. Ochiai, Counting functions for branched covers of elliptic curves and quasi-modular forms, arXiv:math-ph/9909023, 1999.
S. Ramanujan, On the coefficients in the expansions of certain modular functions, Proc. Royal Soc., A, 95 (1918), 144-155.
Michael Somos, Introduction to Ramanujan theta functions
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Seven Staggering Sequences.
Maryna S. Viazovska, The sphere packing problem in dimension 8, arXiv preprint arXiv:1603.04246 [math.NT], 2016.
Martin H. Weissman, Octonions, Cubes, Embeddings, March 2, 2009.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Eisenstein Series.
Eric Weisstein's World of Mathematics, Leech Lattice.
Eric Weisstein's World of Mathematics, Barnes-Wall Lattice
Wikipedia, Eisenstein series
Wikipedia, E_8 lattice
Index entries for sequences related to Eisenstein series
Index entries for sequences related to Barnes-Wall lattices

Cf. A046948 (partial sums), A000143, A108091 (eighth root).
Cf. A008410, A008411, A001158.
Cf. A006352 (E_2), A013973 (E_6), A008410 (E_8), A013974 (E_10), A029828 (E_12), A058550 (E_14), A029829 (E_16), A029830 (E_20), A029831 (E_24).
Cf. A007331 (theta_2(q)^8 / 256), A000143 (theta_3(q)^8), A035016 (theta_4(q)^8).

Basis( ModularForms( Gamma1(1), 4), 29) [1]; /* Michael Somos, May 11 2015 */

L := Lattice("E",8); A := ThetaSeries(L, 57); A; /* Michael Somos, Jun 10 2019 */

with(numtheory); E := proc(k) local n,t1; t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n,n=1..60); series(t1,q,60); end; E(4);

a[ n_] := If[ n < 1, Boole[n == 0], 240 DivisorSigma[ 3, n]]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, t2^2 + 14 t2 t3 + t3^2], {q, 0, n}]; (* Michael Somos, Jun 04 2014 *)
max = 30; s = 1 + 240*Sum[k^3*(q^k/(1 - q^k)), {k, 1, max}] + O[q]^max; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, after Gene Ward Smith *)
a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, t2^2 - t2 t3 + t3^2], {q, 0, 2 n}]; (* Michael Somos, Jul 31 2016 *)

{a(n) = if( n<1, n==0, 240 * sigma(n, 3))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^24 + 256 * x * eta(x^2 + A)^24) / (eta(x + A) * eta(x^2 + A))^8, n))}; /* Michael Somos, Dec 30 2008 */

q='q+O('q^50); Vec((eta(q)^24+256*q*eta(q^2)^24)/(eta(q)*eta(q^2))^8) \\ Altug Alkan, Sep 30 2018

from sympy import divisor_sigma
def a(n): return 1 if n == 0 else 240 * divisor_sigma(n, 3)
[a(n) for n in range(51)]  # Indranil Ghosh, Jul 15 2017

ModularForms(Gamma1(1), 4, prec=30).0 ; # Michael Somos, Jun 04 2013

Can also be expressed as E4(q) = 1 + 240*Sum_{i >= 1} i^3 q^i/(1 - q^i) - Gene Ward Smith, Aug 22 2006
Theta series of E_8 lattice = 1 + 240 * Sum_{m >= 1} sigma_3(m) * q^(2*m), where sigma_3(m) is the sum of the cubes of the divisors of m (A001158).
Expansion of (phi(-q)^8 - (2 * phi(-q) * phi(q))^4 + 16 * phi(q)^8) in powers of q where phi() is a Ramanujan theta function.
Expansion of (eta(q)^24 + 256 * eta(q^2)^24) / (eta(q) * eta(q^2))^8 in powers of q. - Michael Somos, Dec 30 2008
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 + 33*v^2 + 256*w^2 - 18*u*v + 16*u*w - 288*v*w . - Michael Somos, Jan 05 2006
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^2 + 16*u2^2 + 81*u3^2 + 1296*u6^2 - 14*u1*u2 - 18*u1*u3 + 30*u1*u6 + 30*u2*u3 - 288*u2*u6 - 1134*u3*u6 . - Michael Somos, Apr 15 2007
G.f. A(x) satisfies 0 = f(A(x), A(x^3), A(x^9)) where f(u, v, w) = u^3*v + 9*w*u^3 - 84*u^2*v^2 + 246*u*v^3 - 253*v^4 - 675*w*u^2*v + 729*w^2*u^2 - 4590*w*u*v^2 + 19926*w*v^3 - 54675*w^2*u*v + 59049*w^3*u + 531441*w^3*v - 551124*w^2*v^2 . - Michael Somos, Apr 15 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / t) = (t/i)^4 * f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 30 2008
Convolution square is A008410. A008411 is convolution of this sequence with A008410.
Expansion of Ramanujan's function Q(q^2) = 12 (omega/Pi)^4 g2 (Weierstrass invariant) in powers of q^2.
Expansion of a(q) * (a(q)^3 + 8*c(q)^3) in powers of q where a(), c() are cubic AGM theta functions. - Michael Somos, Jan 14 2015
G.f. is (theta_2(q)^8 + theta_3(q)^8 + theta_4(q)^8) / 2 where q = exp(Pi i t). So a(n) = A008430(n) + 128*A007331(n) (= A000143(2*n) + 128*A007331(n) = A035016(2*n) + 128*A007331(n)). - Seiichi Manyama, Sep 30 2018
a(n) = 240*A001158(n) if n>0. - Michael Somos, Oct 01 2018
Sum_{k=1..n} a(k) ~ 2 * Pi^4 * n^4 / 3. - Vaclav Kotesovec, Jan 14 2024

A015128 Number of overpartitions of n: an overpartition of n is an ordered sequence of nonincreasing integers that sum to n, where the first occurrence of each integer may be overlined.

Original entry on oeis.org

1, 2, 4, 8, 14, 24, 40, 64, 100, 154, 232, 344, 504, 728, 1040, 1472, 2062, 2864, 3948, 5400, 7336, 9904, 13288, 17728, 23528, 31066, 40824, 53408, 69568, 90248, 116624, 150144, 192612, 246256, 313808, 398640, 504886, 637592, 802936, 1008448

Offset: 0

N. J. A. Sloane

The over-partition function.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Also the number of jagged partitions of n.
According to Ramanujan (1913) a(n) is close to (cosh(x)-sinh(x)/x)/(4*n) where x=Pi*sqrt(n). - Michael Somos, Mar 17 2003
Number of partitions of 2n with all odd parts occurring with even multiplicities. There is no restriction on the even parts. Cf. A006950, A046682. - Mamuka Jibladze, Sep 05 2003
Number of partitions of n where there are two kinds of odd parts. - Joerg Arndt, Jul 30 2011. Or, in Gosper's words, partitions into red integers and blue odd integers. - N. J. A. Sloane, Jul 04 2016.
Coincides with the sequence of numbers of nilpotent conjugacy classes in the Lie algebras sp(n), n=0,1,2,3,... (the case n=0 being degenerate). A006950, this sequence and A000041 together cover the nilpotent conjugacy classes in the classical A,B,C,D series of Lie algebras. - Alexander Elashvili, Sep 08 2003
Also, number of 01-partitions of n. A 01-partition of n is a weakly decreasing sequence of m nonnegative integers n(i) such that sum(n(i))=n, n(m)>0, n(j)>=n(j+1)-1 and n(j)>=n(j+2). They are special cases of jagged partitions.
a(8n+7) is divisible by 64 (from Fortin/Jacob/Mathieu paper).
Smallest sequence of even numbers (except a(0)) which is the Euler transform of a sequence of positive integers. - Franklin T. Adams-Watters, Oct 16 2006
Convolution of A000041 and A000009. - Vladeta Jovovic, Nov 26 2002
Equals A022567 convolved with A035363. - Gary W. Adamson, Jun 09 2009
Equals the infinite product [1,2,2,2,...] * [1,0,2,0,2,0,2,...] * [1,0,0,2,0,0,2,0,0,2,...] * ... . - Gary W. Adamson, Jul 05 2009
Equals A182818 convolved with A010815. - Gary W. Adamson, Jul 20 2012
Partial sums of A211971. - Omar E. Pol, Jan 09 2014
Also 1 together with the row sums of A235790. - Omar E. Pol, Jan 19 2014
Antidiagonal sums of A284592. - Peter Bala, Mar 30 2017
The overlining method is equivalent to enumerating the k-subsets of the distinct parts of the i-th partition. - Richard Joseph Boland, Sep 02 2021

			G.f. = 1 + 2*q + 4*q^2 + 8*q^3 + 14*q^4 + 24*q^5 + 40*q^6 + 64*q^7 + 100*q^8 + ...
For n = 4 the 14 overpartitions of 4 are [4], [4'], [2, 2], [2', 2], [3, 1], [3', 1], [3, 1'], [3', 1'], [2, 1, 1], [2', 1, 1], [2, 1', 1], [2', 1', 1], [1, 1, 1, 1], [1', 1, 1, 1]. - _Omar E. Pol_, Jan 19 2014

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 103.
R. W. Gosper, Experiments and discoveries in q-trigonometry, in Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Editors: F. G. Garvan and M. E. H. Ismail. Kluwer, Dordrecht, Netherlands, 2001, pp. 79-105. See the function g(q).
James R. Newman, The World of Mathematics, Simon and Schuster, 1956, Vol. I p. 372.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)
Gert Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
Brennan Benfield and Arindam Roy, Log-concavity And The Multiplicative Properties of Restricted Partition Functions, arXiv:2404.03153 [math.NT], 2024.
Noureddine Chair, Partition identities from Partial Supersymmetry, arXiv:hep-th/0409011, 2004.
Shi-Chao Chen, On the number of overpartitions into odd parts, Discrete Math. 325 (2014), 32--37. MR3181230.
William Y.C. Chen and Ernest X.W. Xia, Proof of a conjecture of Hirschhorn and Sellers on overpartitions, arXiv:1307.4155 [math.CO], 2013; Acta Arith. 163 (2014), no. 1, 59--69. MR3194057
Sylvie Corteel, Particle seas and basic hypergeometric series, Advances Appl. Math., 31 (2003), 199-214.
Sylvie Corteel and Jeremy Lovejoy, Frobenius partitions and the combinatorics of Ramanujan's 1 psi 1 summation, J. Combin. Theory A 97 (2002), 177-183.
Sylvie Corteel and Jeremy Lovejoy, Overpartitions, Trans. Amer. Math. Soc., 356 (2004), 1623-1635.
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
Benjamin Engel, Log-concavity of the overpartition function, The Ramanujan Journal, Vol. 43, No. 2 (2017), pp. 229-241; arXiv preprint, arXiv:1412.4603 [math.NT], 2014.
Alex Fink, Richard K. Guy and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008).
J.-F. Fortin, P. Jacob and P. Mathieu, Jagged partitions, The Ramanujan Journal, Vol. 10, No. 2 (2005), pp. 215-235; arXiv preprint, arXiv:math/0310079 [math.CO], 2003-2005.
Frank Garvan, Table of a(n) for n = 1..10000.
R. W. Gosper, Experiments and discoveries in q-trigonometry, in F. G. Garvan and M. E. H. Ismail (eds.), Symbolic computation, number theory, special functions, physics and combinatorics, Springer, Boston, MA, 2001, pp. 79-105; preprint.
R. W. Gosper, q-Trigonometry: Some Prefatory Afterthoughts
William J. Keith, Restricted k-color partitions, The Ramanujan Journal, Vol. 40, No. 1 (2016), pp. 71-92; arXiv preprint, arXiv:1408.4089 [math.CO], 2014.
Byungchan Kim, A short note on the overpartition function, Discr. Math., 309 (2009), 2528-2532.
Byungchan Kim, Overpartition pairs modulo powers of 2, Discrete Math., 311 (2011), 835-840.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
Jeremy Lovejoy, Gordon's theorem for overpartitions, J. Combin. Theory A 103 (2003), 393-401.
Karl Mahlburg, The overpartition function modulo small powers of 2, Discr. Math., 286 (2004), 263-267.
Mircea Merca, Overpartitions in terms of 2-adic valuation, Aequat. Math. (2024). See p. 9.
Igor Pak, Partition bijections, a survey, The Ramanujan Journal, Vol. 12, No. 1 (2006), pp. 5-75; alternative link.
Maxie D. Schmidt, Exact Formulas for the Generalized Sum-of-Divisors Functions, arXiv:1705.03488 [math.NT], 2017-2019. See Example 4.1, p. 13.
Michael Somos, Introduction to Ramanujan theta functions
Liuquan Wang, Another Proof of a Conjecture by Hirschhorn and Sellers on Overpartitions, J. Int. Seq. 17 (2014) # 14.9.8.
Eric Weisstein's World of Mathematics, Modular Equation.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Michael P. Zaletel and Roger S. K. Mong, Exact matrix product states for quantum Hall wave functions, Physical Review B, Vol. 86, No. 24 (2012), 245305; arXiv preprint, arXiv:1208.4862 [cond-mat.str-el], 2012. - From _N. J. A. Sloane_, Dec 25 2012

Cf. A022567, A035363, A002448, A106507, A156616, A261386, A265835, A014968, A284592, A303662.
See A004402 for a version with signs.
Column k=2 of A321884.
Cf. A002513.

# JacobiTheta4 is defined in A002448.
A015128List(len) = JacobiTheta4(len, -1)
A015128List(40) |> println # Peter Luschny, Mar 12 2018

mul((1+x^n)/(1-x^n),n=1..256): seq(coeff(series(%,x,n+1),x,n), n=0..40);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +2*add(b(n-i*j, i-1), j=1..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 10 2014
a_list := proc(len) series(1/JacobiTheta4(0,x),x,len+1); seq(coeff(%,x,j),j=0..len) end: a_list(39); # Peter Luschny, Mar 14 2017

max = 39; f[x_] := Exp[Sum[(DivisorSigma[1, 2*n] - DivisorSigma[1, n])*(x^n/n), {n, 1, max}]]; CoefficientList[ Series[f[x], {x, 0, max}], x] (* Jean-François Alcover, Jun 11 2012, after Joerg Arndt *)
a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {-1}, {}, x, x], {x, 0, n}]; (* Michael Somos, Mar 11 2014 *)
QP = QPochhammer; s = QP[q^2]/QP[q]^2 + O[q]^40; CoefficientList[s + O[q]^100, q] (* Jean-François Alcover, Nov 25 2015, after Michael Somos *)
Table[Sum[PartitionsP[n-k]*PartitionsQ[k], {k, 0, n}], {n, 0, 50}] (* Vaclav Kotesovec, Nov 28 2015 *)
(QPochhammer[-x, x]/QPochhammer[x, x] + O[x]^50)[[3]] (* Vladimir Reshetnikov, Nov 12 2016 *)
nmax = 100; p = ConstantArray[0, nmax+1]; p[[1]] = 1; Do[p[[n+1]] = 0; k = 1; While[n + 1 - k^2 > 0, p[[n+1]] += (-1)^(k+1)*p[[n + 1 - k^2]]; k++;]; p[[n+1]] = 2*p[[n+1]];, {n, 1, nmax}]; p (* Vaclav Kotesovec, Apr 11 2017 *)
a[ n_] := SeriesCoefficient[ 1 / EllipticTheta[ 4, 0, x], {x, 0, n}]; (* Michael Somos, Nov 15 2018 *)
a[n_] := Sum[2^Length[Union[IntegerPartitions[n][[i]]]], {i, 1, PartitionsP[n]}]; (* Richard Joseph Boland, Sep 02 2021 *)
n = 39; CoefficientList[Product[(1 + x^k)/(1 - x^k), {k, 1, n}] + O[x]^(n + 1), x] (* Oliver Seipel, Sep 19 2021 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) / eta(x + A)^2, n))}; /* Michael Somos, Nov 01 2008 */

{a(n)=polcoeff(exp(sum(m=1,n\2+1,2*x^(2*m-1)/(1-x^(2*m-1)+x*O(x^n))/(2*m-1))),n)} /* Paul D. Hanna, Aug 06 2009 */

N=66; x='x+O('x^N); gf=exp(sum(n=1,N,(sigma(2*n)-sigma(n))*x^n/n));Vec(gf) /* Joerg Arndt, Jul 30 2011 */

lista(nn) = {q='q+O('q^nn); Vec(eta(q^2)/eta(q)^2)} \\ Altug Alkan, Mar 20 2018

# uses[EulerTransform from A166861]
a = BinaryRecurrenceSequence(0, 1, 1, 2)
b = EulerTransform(a)
print([b(n) for n in range(40)]) # Peter Luschny, Nov 11 2020

Euler transform of period 2 sequence [2, 1, ...]. - Michael Somos, Mar 17 2003
G.f.: Product_{m>=1} (1 + q^m)/(1 - q^m).
G.f.: 1 / (Sum_{m=-inf..inf} (-q)^(m^2)) = 1/theta_4(q).
G.f.: 1 / Product_{m>=1} (1 - q^(2*m)) * (1 - q^(2*m-1))^2.
G.f.: exp( Sum_{n>=1} 2*x^(2*n-1)/(1 - x^(2*n-1))/(2*n-1) ). - Paul D. Hanna, Aug 06 2009
G.f.: exp( Sum_{n>=1} (sigma(2*n) - sigma(n))*x^n/n ). - Joerg Arndt, Jul 30 2011
G.f.: Product_{n>=0} theta_3(q^(2^n))^(2^n). - Joerg Arndt, Aug 03 2011
A004402(n) = (-1)^n * a(n). - Michael Somos, Mar 17 2003
Expansion of eta(q^2) / eta(q)^2 in powers of q. - Michael Somos, Nov 01 2008
Expansion of 1 / phi(-q) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Nov 01 2008
Convolution inverse of A002448. - Michael Somos, Nov 01 2008
Recurrence: a(n) = 2*Sum_{m>=1} (-1)^(m+1) * a(n-m^2).
a(n) = (1/n)*Sum_{k=1..n} (sigma(2*k) - sigma(k))*a(n-k). - Vladeta Jovovic, Dec 05 2004
G.f.: Product_{i>=1} (1 + x^i)^A001511(2i) (see A000041). - Jon Perry, Jun 06 2004
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = w^4 * (u^4 + v^4) - 2 * u^2 * v^6. - Michael Somos, Nov 01 2008
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u6^3 * (u1^2 + u3^2) - 2 * u1 * u2 * u3^3. - Michael Somos, Nov 01 2008
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u2^3 * (u3^2 - 3 * u1^2) + 2 * u1^3 * u3 * u6. - Michael Somos, Nov 01 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 32^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A106507. - Michael Somos, Nov 01 2008
a(n) = 2*A014968(n), n >= 1. - Omar E. Pol, Jan 19 2014
a(n) ~ Pi * BesselI(3/2, Pi*sqrt(n)) / (4*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Jan 11 2017
Let T(n,k) = the number of partitions of n with parts 1 through k of two kinds, T(n,0) = A000041(n), the number of partitions of n. Then a(n) = T(n,0) + T(n-1,1) + T(n-3,2) + T(n-6,3) + T(n-10,4) + T(n-15,5) + ... .   Gregory L. Simay, May 29 2019
For n >= 1, a(n) = Sum_{k>=1} 2^k * A116608(n,k). - Gregory L. Simay, Jun 01 2019
Sum_{n>=1} 1/a(n) = A303662. - Amiram Eldar, Nov 15 2020
a(n) = Sum_{i=1..p(n)} 2^(d(n,i)), where d(n,i) is the number of distinct parts in the i-th partition of n. - Richard Joseph Boland, Sep 02 2021
G.f.: A(x) = exp( Sum_{n >= 1} x^n*(2 + x^n)/(n*(1 - x^(2*n))) ). - Peter Bala, Dec 23 2021
G.f. A(q) satisfies (3*A(q)/A(q^9) - 1)^3 = 9*A(q)^4/A(q^3)^4 - 1. - Paul D. Hanna, Oct 14 2024

Minor edits by Vaclav Kotesovec, Sep 13 2014

A027187 Number of partitions of n into an even number of parts.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 6, 7, 12, 14, 22, 27, 40, 49, 69, 86, 118, 146, 195, 242, 317, 392, 505, 623, 793, 973, 1224, 1498, 1867, 2274, 2811, 3411, 4186, 5059, 6168, 7427, 9005, 10801, 13026, 15572, 18692, 22267, 26613, 31602, 37619, 44533, 52815, 62338, 73680, 86716, 102162, 119918

Offset: 0

Clark Kimberling

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
For n > 0, also the number of partitions of n whose greatest part is even. [Edited by Gus Wiseman, Jan 05 2021]
Number of partitions of n+1 into an odd number of parts, the least being 1.
Also the number of partitions of n such that the number of even parts has the same parity as the number of odd parts; see Comments at A027193. - Clark Kimberling, Feb 01 2014, corrected Jan 06 2021
Suppose that c(0) = 1, that c(1), c(2), ... are indeterminates, that d(0) = 1, and that d(n) = -c(n) - c(n-1)*d(1) - ... - c(0)*d(n-1).  When d(n) is expanded as a polynomial in c(1), c(2),..,c(n), the terms are of the form H*c(i_1)*c(i_2)*...*c(i_k).  Let P(n) = [c(i_1), c(i_2), ..., c(i_k)], a partition of n.  Then H is negative if P has an odd number of parts, and H is positive if P has an even number of parts.  That is, d(n) has A027193(n) negative coefficients, A027187(n) positive coefficients, and A000041 terms.  The maximal coefficient in d(n), in absolute value, is A102462(n). - Clark Kimberling, Dec 15 2016

			G.f. = 1 + x^2 + x^3 + 3*x^4 + 3*x^5 + 6*x^6 + 7*x^7 + 12*x^8 + 14*x^9 + 22*x^10 + ...
From _Gus Wiseman_, Jan 05 2021: (Start)
The a(2) = 1 through a(8) = 12 partitions into an even number of parts are the following. The Heinz numbers of these partitions are given by A028260.
  (11)  (21)  (22)    (32)    (33)      (43)      (44)
              (31)    (41)    (42)      (52)      (53)
              (1111)  (2111)  (51)      (61)      (62)
                              (2211)    (2221)    (71)
                              (3111)    (3211)    (2222)
                              (111111)  (4111)    (3221)
                                        (211111)  (3311)
                                                  (4211)
                                                  (5111)
                                                  (221111)
                                                  (311111)
                                                  (11111111)
The a(2) = 1 through a(8) = 12 partitions whose greatest part is even are the following. The Heinz numbers of these partitions are given by A244990.
  (2)  (21)  (4)    (41)    (6)      (43)      (8)
             (22)   (221)   (42)     (61)      (44)
             (211)  (2111)  (222)    (421)     (62)
                            (411)    (2221)    (422)
                            (2211)   (4111)    (431)
                            (21111)  (22111)   (611)
                                     (211111)  (2222)
                                               (4211)
                                               (22211)
                                               (41111)
                                               (221111)
                                               (2111111)
(End)

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; See p. 8, (7.323) and p. 39, Example 7.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)
George E. Andrews and David Newman, The Minimal Excludant in Integer Partitions, J. Int. Seq., Vol. 23 (2020), Article 20.2.3.
Arvind Ayyer, Hiranya Kishore Dey, and Digjoy Paul, How large is the character degree sum compared to the character table sum for a finite group?, arXiv:2406.06036 [math.RT], 2024. See p. 13.
Roland Bacher and Pierre De La Harpe, Conjugacy growth series of some infinitely generated groups, International Mathematics Research Notices, 2016, pp.1-53. (hal-01285685v2)
N. J. Fine, Problem 4314, Amer. Math. Monthly, Vol. 57, 1950, 421-423.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function p_e(n).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000701, A046682, A026838, A102462.
The Heinz numbers of these partitions are A028260.
The odd version is A027193.
The strict case is A067661.
The case of even sum as well as length is A236913 (the even bisection).
Other cases of even length:
- A024430 counts set partitions of even length.
- A034008 counts compositions of even length.
- A052841 counts ordered set partitions of even length.
- A174725 counts ordered factorizations of even length.
- A332305 counts strict compositions of even length
- A339846 counts factorizations of even length.
A000009 counts partitions into odd parts, ranked by A066208.
A026805 counts partitions whose least part is even.
A072233 counts partitions by sum and length.
A101708 counts partitions of even positive rank.
Cf. A000700, A026424, A058696, A096373, A244990, A300061.

f[n_] := Length[Select[IntegerPartitions[n], IntegerQ[First[#]/2] &]]; Table[f[n], {n, 1, 30}] (* Clark Kimberling, Mar 13 2012 *)
a[ n_] := SeriesCoefficient[ (1 + EllipticTheta[ 4, 0, x]) / (2 QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, May 06 2015 *)
a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[n], EvenQ[Length @ #] &]]; (* Michael Somos, May 06 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum( k=0, sqrtint(n), (-x)^k^2, A) / eta(x + A), n))}; /* Michael Somos, Aug 19 2006 */

my(q='q+O('q^66)); Vec( (1/eta(q)+eta(q)/eta(q^2))/2 ) \\ Joerg Arndt, Mar 23 2014

a(n) = (A000041(n) + (-1)^n * A000700(n))/2.
a(n) = p(n) - p(n-1) + p(n-4) - p(n-9) + ... where p(n) is the number of unrestricted partitions of n, A000041. [Fine] - David Callan, Mar 14 2004
From Bill Gosper, Jun 25 2005: (Start)
G.f.: A(q) = Sum_{n >= 0} a(n) q^n = 1 + q^2 + q^3 + 3*q^4 + 3*q^5 + 6*q^6 + ...
  = Sum_{n >= 0} q^(2*n)/(q; q)_{2*n}
  = ((Product_{k >= 1} 1/(1-q^k)) + (Product_{k >= 1} 1/(1+q^k)))/2.
Also, let B(q) = Sum_{n >= 0} A027193(n) q^n = q + q^2 + 2*q^3 + 2*q^4 + 4*q^5 + 5*q^6 + ...
Then B(q) = Sum_{n >= 0} q^(2*n+1)/(q; q){2*n+1} = ((Product{k >= 1} 1/(1-q^k)) - (Product_{k >= 1} 1/(1+q^k)))/2.
Also we have the following identity involving 2 X 2 matrices:
Product_{k >= 1} [ 1/(1-q^(2*k)), q^k/(1-q^(2*k)) ; q^k/(1-q^(2*k)), 1/(1-q^(2*k)) ]
  = [ A(q), B(q) ; B(q), A(q) ]. (End)
a(2*n) = A046682(2*n), a(2*n+1) = A000701(2*n+1); a(n) = A000041(n)-A027193(n). - Reinhard Zumkeller, Apr 22 2006
Expansion of (1 + phi(-q)) / (2 * f(-q)) where phi(), f() are Ramanujan theta functions. - Michael Somos, Aug 19 2006
G.f.: (Sum_{k>=0} (-1)^k * x^(k^2)) / (Product_{k>0} (1 - x^k)). - Michael Somos, Aug 19 2006
a(n) = A338914(n) + A096373(n). - Gus Wiseman, Jan 06 2021

Offset changed to 0 by Michael Somos, Jul 24 2012

A004018 Theta series of square lattice (or number of ways of writing n as a sum of 2 squares). Often denoted by r(n) or r_2(n).

Original entry on oeis.org

1, 4, 4, 0, 4, 8, 0, 0, 4, 4, 8, 0, 0, 8, 0, 0, 4, 8, 4, 0, 8, 0, 0, 0, 0, 12, 8, 0, 0, 8, 0, 0, 4, 0, 8, 0, 4, 8, 0, 0, 8, 8, 0, 0, 0, 8, 0, 0, 0, 4, 12, 0, 8, 8, 0, 0, 0, 0, 8, 0, 0, 8, 0, 0, 4, 16, 0, 0, 8, 0, 0, 0, 4, 8, 8, 0, 0, 0, 0, 0, 8, 4, 8, 0, 0, 16, 0, 0, 0, 8, 8, 0, 0, 0, 0, 0, 0, 8, 4, 0, 12, 8

Offset: 0

N. J. A. Sloane

Number of points in square lattice on the circle of radius sqrt(n). Equivalently, number of Gaussian integers of norm n (cf. Conway-Sloane, p. 106).
Let b(n)=A004403(n), then Sum_{k=1..n} a(k)*b(n-k) = 1. - John W. Layman
Theta series of D_2 lattice.
Number 6 of the 74 eta-quotients listed in Table I of Martin (1996).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
The zeros in this sequence correspond to those integers with an equal number of 4k+1 and 4k+3 divisors, or equivalently to those that have at least one 4k+3 prime factor with an odd exponent (A022544). - Ant King, Mar 12 2013
If A(q) = 1 + 4*q + 4*q^2 + 4*q^4 + 8*q^5 + ... denotes the o.g.f. of this sequence then the function F(q) := 1/4*(A(q^2) - A(q^4)) = ( Sum_{n >= 0} q^(2*n+1)^2 )^2 is the o.g.f. for counting the ways a positive integer n can be written as the sum of two positive odd squares. - Peter Bala, Dec 13 2013
Expansion coefficients of (2/Pi)*K, with the real quarter period K of elliptic functions, as series of the Jacobi nome q, due to (2/Pi)*K = theta_3(0,q)^2. See, e.g., Whittaker-Watson, p. 486. - Wolfdieter Lang, Jul 15 2016
Sum_{k=0..n} a(n) = A057655(n). Robert G. Wilson v, Dec 22 2016
Limit_{n->oo} (a(n)/n - Pi*log(n)) = A062089: Sierpinski's constant. - Robert G. Wilson v, Dec 22 2016
The mean value of a(n) is Pi, see A057655 for more details. - M. F. Hasler, Mar 20 2017

			G.f. = 1 + 4*q + 4*q^2 + 4*q^4 + 8*q^5 + 4*q^8 + 4*q^9 + 8*q^10 + 8*q^13 + 4*q^16 + 8*q^17 + 4*q^18 + 8*q^20 + 12*q^25 + 8*q^26 + ... . - _John Cannon_, Dec 30 2006

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16 (7), r(n).
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.23).
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 15, p. 32, Lemma 2 (with the proof), p. 116, (9.10) first formula.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 133.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 240, r(n).
W. König and J. Sprekels, Karl Weierstraß (1815-1897), Springer Spektrum, Wiesbaden, 2016, p. 186-187 and p. 280-281.
C. D. Olds, A. Lax and G. P. Davidoff, The Geometry of Numbers, Math. Assoc. Amer., 2000, p. 51.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 244-245.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, fourth edition, reprinted, 1958, Cambridge at the University Press.

T. D. Noe, Table of n, a(n) for n = 0..10000
G. E. Andrews, R. Lewis and Z.-G. Liu, An identity relating a theta series to a sum of Lambert series, Bull. London Math. Soc., 33 (2001), 25-31.
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
S. Cooper and M. Hirschhorn, A combinatorial proof of a result from number theory, Integers 4 (2004), #A09.
Michael Gilleland, Some Self-Similar Integer Sequences
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62. (This sequence is called rho, see page 6)
M. D. Hirschhorn, Jacobi's Two-Square Theorem and Related Identities
M. D. Hirschhorn, Arithmetic Consequences of Jacobi's Two-Squares Theorem
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
Jacobi-Legendre letters, Correspondance mathématique entre Legendre et Jacobi, J. Reine Angew. Math. 80 (1875) 205-279, letter of September 9, 1828, p. 240-243, formula for 2K/Pi on p. 242.
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015. [Note that a later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication which is next in this list.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016.
Christian Kassel, Christophe Reutenauer, The Fourier expansion of eta(z)eta(2z)eta(3z)/eta(6z), arXiv:1603.06357 [math.NT], 2016.
M. Kontsevich and D. Zagier, Periods, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22. Published in B. Engquist and W. Schmid, editors, Mathematics Unlimited - 2001 and Beyond, 2 vols., Springer-Verlag, 2001, pp. 771-808, section 2.3. Example 3.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
G. Nebe and N. J. A. Sloane, Home page for this lattice
F. Richman, Counting Gaussian integers in a disk
Grant Sanderson, Pi hiding in prime regularities, 3Blue1Brown video (2017).
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
G. Villemin, Sommes de 2 carrés
Eric Weisstein's World of Mathematics, Barnes-Wall Lattice
Eric Weisstein's World of Mathematics, Moebius Transform
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Sum of Squares Function
Eric Weisstein's World of Mathematics, Theta Series
Zach Wissner-Gross a.k.a. The Riddler, Riddler Express: Can You Climb Your Way To Victory?, FiveThirtyEight, Sep 24 2021.
G. Xiao, Two squares
Index entries for sequences related to sums of squares
Index entries for sequences mentioned by Glaisher
Index entries for "core" sequences

Row d=2 of A122141 and of A319574, 2nd column of A286815.
Partial sums - 1 give A014198.
A071385 gives records; A071383 gives where records occur.
Cf. A001481, A004020, A005883, A057655 (partial sums), A057961, A057962, A002654,
A104271, A105673.

# JacobiTheta3 is defined in A000122.
A004018List(len) = JacobiTheta3(len, 2)
A004018List(102) |> println # Peter Luschny, Mar 12 2018

Basis( ModularForms( Gamma1(4), 1), 100) [1]; /* Michael Somos, Jun 10 2014 */

(sum(x^(m^2),m=-10..10))^2;
# Alternative:
A004018list := proc(len) series(JacobiTheta3(0, x)^2, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end:
t1 := A004018list(102);
r2 := n -> t1[n+1]; # Peter Luschny, Oct 02 2018

SquaresR[2,Range[0,110]] (* Harvey P. Dale, Oct 10 2011 *)
a[ n_] := SquaresR[ 2, n]; (* Michael Somos, Nov 15 2011 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^2, {q, 0, n}]; (* Michael Somos, Nov 15 2011 *)
a[ n_] := With[{m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ EllipticK[ m] / (Pi/2), {q, 0, n}]]; (* Michael Somos, Jun 10 2014 *)
a[ n_] := If[ n < 1, Boole[n == 0], 4 Sum[ KroneckerSymbol[-4, d], {d, Divisors@n}]]; (* or *) a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^10/(QPochhammer[ q] QPochhammer[ q^4])^4, {q, 0, n}]; (* Michael Somos, May 17 2015 *)

{a(n) = polcoeff( 1 + 4 * sum( k=1, n, x^k / (1 + x^(2*k)), x * O(x^n)), n)}; /* Michael Somos, Mar 14 2003 */

{a(n) = if( n<1, n==0, 4 * sumdiv( n, d, (d%4==1) - (d%4==3)))}; /* Michael Somos, Jul 19 2004 */

{a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0; 0, 1], n)[n])}; /* Michael Somos, May 13 2005 */

a(n)=if(n==0,return(1)); my(f=factor(n)); 4*prod(i=1,#f~, if(f[i,1]%4==1, f[i,2]+1, if(f[i,2]%2 && f[i,1]>2, 0, 1))) \\ Charles R Greathouse IV, Sep 02 2015

from sympy import factorint
def a(n):
    if n == 0: return 1
    an = 4
    for pi, ei in factorint(n).items():
       if pi%4 == 1: an *= ei+1
       elif pi%4 == 3 and ei%2: return 0
    return an
print([a(n) for n in range(102)]) # Michael S. Branicky, Sep 24 2021

from math import prod
from sympy import factorint
def A004018(n): return prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in factorint(n).items())<<2 if n else 1 # Chai Wah Wu, Jul 07 2022, corrected Jun 21 2024.

Q = DiagonalQuadraticForm(ZZ, [1]*2)
Q.representation_number_list(102) # Peter Luschny, Jun 20 2014

Expansion of theta_3(q)^2 = (Sum_{n=-oo..+oo} q^(n^2))^2 = Product_{m>=1} (1-q^(2*m))^2 * (1+q^(2*m-1))^4; convolution square of A000122.
Factor n as n = p1^a1 * p2^a2 * ... * q1^b1 * q2^b2 * ... * 2^c, where the p's are primes == 1 (mod 4) and the q's are primes == 3 (mod 4). Then a(n) = 0 if any b is odd, otherwise a(n) = 4*(1 + a1)*(1 + a2)*...
G.f. = s(2)^10/(s(1)^4*s(4)^4), where s(k) := subs(q=q^k, eta(q)) and eta(q) is Dedekind's function, cf. A010815. [Fine]
a(n) = 4*A002654(n), n > 0.
Expansion of eta(q^2)^10 / (eta(q) * eta(q^4))^4 in powers of q. - Michael Somos, Jul 19 2004
Expansion of ( phi(q)^2 + phi(-q)^2 ) / 2 in powers of q^2 where phi() is a Ramanujan theta function.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u - v)^2 - (v - w) * 4 * w. - Michael Somos, Jul 19 2004
Euler transform of period 4 sequence [4, -6, 4, -2, ...]. - Michael Somos, Jul 19 2004
Moebius transform is period 4 sequence [4, 0, -4, 0, ...]. - Michael Somos, Sep 17 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2 (t/i) f(t) where q = exp(2 Pi i t).
The constant sqrt(Pi)/Gamma(3/4)^2 produces the first 324 terms of the sequence when expanded in base exp(Pi), 450 digits of the constant are necessary. - Simon Plouffe, Mar 03 2011
a(n) = A004531(4*n). a(n) = 2*A105673(n), if n>0.
Let s = 16*q*(E1*E4^2/E2^3)^8 where Ek = Product_{n>=1} (1-q^(k*n)) (s=k^2 where k is elliptic k), then the g.f. is hypergeom([+1/2, +1/2], [+1], s) (expansion of 2/Pi*ellipticK(k) in powers of q). - Joerg Arndt, Aug 15 2011
Dirichlet g.f. Sum_{n>=1} a(n)/n^s = 4*zeta(s)*L_(-4)(s), where L is the D.g.f. of the (shifted) A056594. [Raman. J. 7 (2003) 95-127]. - R. J. Mathar, Jul 02 2012
a(n) = floor(1/(n+1)) + 4*floor(cos(Pi*sqrt(n))^2) - 4*floor(cos(Pi*sqrt(n/2))^2) + 8*Sum_{i=1..floor(n/2)} floor(cos(Pi*sqrt(i))^2)*floor(cos(Pi*sqrt(n-i))^2). - Wesley Ivan Hurt, Jan 09 2013
From Wolfdieter Lang, Aug 01 2016: (Start)
A Jacobi identity: theta_3(0, q)^2 = 1 + 4*Sum_{r>=0} (-1)^r*q^(2*r+1)/(1 - q^(2*r+1)). See, e.g., the Grosswald reference (p. 15, p. 116, but p. 32, Lemma 2 with the proof, has the typo r >= 1 instead of r >= 0 in the sum, also in the proof). See the link with the Jacobi-Legendre letter.
Identity used by Weierstraß (see the König-Sprekels book, p. 187, eq. (5.12) and p. 281, with references, but there F(x) from (5.11) on p. 186 should start with nu =1 not 0): theta_3(0, q)^2 = 1 + 4*Sum_{n>=1} q^n/(1 + q^(2*n)). Proof: similar to the one of the preceding Jacobi identity. (End)
a(n) = (4/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017
G.f.: Theta_3(q)^2 = hypergeometric([1/2, 1/2],[1],lambda(q)), with lambda(q) = Sum_{j>=1} A115977(j)*q^j. See the Kontsevich and Zagier link, with Theta -> Theta_3, z -> 2*z and q -> q^2. - Wolfdieter Lang, May 27 2018

A002878 Bisection of Lucas sequence: a(n) = L(2*n+1).

Original entry on oeis.org

1, 4, 11, 29, 76, 199, 521, 1364, 3571, 9349, 24476, 64079, 167761, 439204, 1149851, 3010349, 7881196, 20633239, 54018521, 141422324, 370248451, 969323029, 2537720636, 6643838879, 17393796001, 45537549124, 119218851371, 312119004989, 817138163596, 2139295485799

Offset: 0

N. J. A. Sloane

In any generalized Fibonacci sequence {f(i)}, Sum_{i=0..4n+1} f(i) = a(n)*f(2n+2). - Lekraj Beedassy, Dec 31 2002
The continued fraction expansion for F((2n+1)*(k+1))/F((2n+1)*k), k>=1 is [a(n),a(n),...,a(n)] where there are exactly k elements (F(n) denotes the n-th Fibonacci number). E.g., continued fraction for F(12)/F(9) is [4, 4,4]. - Benoit Cloitre, Apr 10 2003
See A135064 for a possible connection with Galois groups of quintics.
Sequence of all positive integers k such that continued fraction [k,k,k,k,k,k,...] belongs to Q(sqrt(5)). - Thomas Baruchel, Sep 15 2003
All positive integer solutions of Pell equation a(n)^2 - 5*b(n)^2 = -4 together with b(n)=A001519(n), n>=0.
a(n) = L(n,-3)*(-1)^n, where L is defined as in A108299; see also A001519 for L(n,+3).
Inverse binomial transform of A030191. - Philippe Deléham, Oct 04 2005
General recurrence is a(n) = (a(1)-1)*a(n-1) - a(n-2), a(1) >= 4, lim_{n->infinity} a(n) =  x*(k*x+1)^n, k =(a(1)-3), x=(1+sqrt((a(1)+1)/(a(1)-3)))/2. Examples in OEIS: a(1)=4 gives A002878. a(1)=5 gives A001834. a(1)=6 gives A030221. a(1)=7 gives A002315. a(1)=8 gives A033890. a(1)=9 gives A057080. a(1)=10 gives A057081. - Ctibor O. Zizka, Sep 02 2008
Let r = (2n+1), then a(n), n>0 = Product_{k=1..floor((r-1)/2)} (1 + sin^2 k*Pi/r); e.g., a(3) = 29 = (3.4450418679...)*(4.801937735...)*(1.753020396...). - Gary W. Adamson, Nov 26 2008
a(n+1) is the Hankel transform of A001700(n)+A001700(n+1). - Paul Barry, Apr 21 2009
a(n) is equal to the permanent of the (2n) X (2n) tridiagonal matrix with sqrt(5)'s along the main diagonal, i's along the superdiagonal and the subdiagonal (i is the imaginary unit), and 0's everywhere else. - John M. Campbell, Jun 09 2011
Conjecture: for n > 0, a(n) = sqrt(Fibonacci(4*n+3) + Sum_{k=2..2*n} Fibonacci(2*k)). - Alex Ratushnyak, May 06 2012
Pisano period lengths: 1, 3, 4, 3, 2, 12, 8, 6, 12, 6, 5, 12, 14, 24, 4, 12, 18, 12, 9, 6, ... . - R. J. Mathar, Aug 10 2012
The continued fraction [a(n); a(n), a(n), ...] = phi^(2n+1), where phi is the golden ratio, A001622. - Thomas Ordowski, Jun 05 2013
Solutions (x, y) = (a(n), a(n+1)) satisfying  x^2 + y^2 = 3xy + 5. - Michel Lagneau, Feb 01 2014
Conjecture: except for the number 3, a(n) are the numbers such that a(n)^2+2 are Lucas numbers. - Michel Lagneau, Jul 22 2014
Comment on the preceding conjecture: It is clear that all a(n) satisfy a(n)^2 + 2 = L(2*(2*n+1)) due to the identity (17 c) of Vajda, p. 177: L(2*n) + 2*(-1)^n = L(n)^2 (take n -> 2*n+1). - Wolfdieter Lang, Oct 10 2014
Limit_{n->oo} a(n+1)/a(n) = phi^2 = phi + 1 = (3+sqrt(5))/2. - Derek Orr, Jun 18 2015
If d[k] denotes the sequence of k-th differences of this sequence, then d[0](0), d[1](1), d[2](2), d[3](3), ... = A048876, cf. message to SeqFan list by P. Curtz on March 2, 2016. - M. F. Hasler, Mar 03 2016
a(n-1) and a(n) are the least phi-antipalindromic numbers (A178482) with 2*n and 2*n+1 digits in base phi, respectively. - Amiram Eldar, Jul 07 2021
Triangulate (hyperbolic) 2-space such that around every vertex exactly 7 triangles touch. Call any 7 triangles having a common vertex the first layer and let the (n+1)-st layer be all triangles that do not appear in any of the first n layers and have a common vertex with the n-th layer. Then the n-th layer contains 7*a(n-1) triangles. E.g., the first layer (by definition) contains 7 triangles, the second layer (the "ring" of triangles around the first layer) consists of 28 triangles, the third layer (the next "ring") consists of 77 triangles, and so on. - Nicolas Nagel, Aug 13 2022

			G.f. = 1 + 4*x + 11*x^2 + 29*x^3 + 76*x^4 + 199*x^5 + 521*x^6 + ... - _Michael Somos_, Jan 13 2019

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 91.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Steven Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Marco Abrate, Stefano Barbero, Umberto Cerruti and Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.
Kasper K. S. Andersen, Lisa Carbone and Diego Penta, Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
Paul Barry, On the Central Coefficients of Bell Matrices, J. Int. Seq., Vol. 14 (2011), Article 11.4.3, page 9.
Hacène Belbachir, Soumeya Merwa Tebtoub and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Joshua P. Bowman, Compositions with an Odd Number of Parts, and Other Congruences, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 25.
Nathan D. Cahill, John R. D'Errico and John P. Spence, Complex Factorizations of the Fibonacci and Lucas Numbers, Fibonacci Quarterly, 1(41):13-19, 2003.
L. Carlitz, Problem B-110, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 5, No. 1 (1967), p. 108; An Infinite Series Equality, Solution to Problem B-110 by the proposer, ibid., Vol. 5, No. 5 (1967), pp. 469-470.
Paul Curtz, A269500, SeqFan list, March 2, 2016.
Murray Elder and Arkadius Kalka, Logspace computations for rigid Garside groups, arXiv preprint arXiv:1310.0933 [math.GR], 2013.
Sergio Falcon, Relationships between Some k-Fibonacci Sequences, Applied Mathematics, Vol. 5 (2014), pp. 2226-2234.
Alex Fink, Richard K. Guy and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol. 3, No. 2 (2008), pp. 76-114. See Section 13.
Dale Gerdemann, Collision of Digits "Also interesting are the two bisections of the Lucas numbers A005248 (digit minimizer) and A002878 (digit maximizer). I particularly like the multiples of A005248 because I have this image of the two digits piling up on top of each other and then spreading out like waves".
André Gougenheim, About the linear sequence of integers such that each term is the sum of the two preceding Part 1 Part 2, Fib. Quart., Vol. 9, No. 3 (1971), pp. 277-295, 298.
Tian-Xiao He and Louis W. Shapiro, Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group, Lin. Alg. Applic., Vol. 532 (2017) pp. 25-41, example p 34.
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 13.
Seong Ju Kim, Ryan Stees and Laura Taalman, Sequences of Spiral Knot Determinants, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.4.
Tanya Khovanova, Recursive Sequences.
Ioana-Claudia Lazăr, Lucas sequences in t-uniform simplicial complexes, arXiv:1904.06555 [math.GR], 2019.
D. H. Lehmer, Recurrence formulas for certain divisor functions, Bull. Amer. Math. Soc., Vol. 49, No. 2 (1943), pp. 150-156.
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum, Vol. 19 (2019), pp. 11-16.
Donatella Merlini and Renzo Sprugnoli, Arithmetic into geometric progressions through Riordan arrays, Discrete Mathematics, Vol. 340, No. 2 (2017), pp. 160-174.
Prabha Sivaraman Nair and Rejikumar Karunakaran, On k-Fibonacci Brousseau Sums, J. Int. Seq. (2024) Art. No. 24.6.4. See p. 3.
Serge Perrine, Some properties of the equation x^2=5y^2-4, The Fibonacci Quarterly, Vol. 54, No. 2 (2016) pp. 172-177.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Ryan Stees, Sequences of Spiral Knot Determinants, Senior Honors Projects, Paper 84, James Madison Univ., May 2016.
Eric Weisstein's World of Mathematics, Fibonacci Polynomial.
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory, Vol. 7, No. 5 (2011), pp. 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences, Integers, Volume 12A (2012), The John Selfridge Memorial Volume.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (3,-1).

Cf. A000204. a(n) = A060923(n, 0), a(n)^2 = A081071(n).
Cf. A005248 [L(2n) = bisection (even n) of Lucas sequence].
Cf. A001906 [F(2n) = bisection (even n) of Fibonacci sequence], A000045, A002315, A004146, A029907, A113224, A153387, A153416, A178482, A192425, A285992 (prime subsequence).
Cf. similar sequences of the type k*F(n)*F(n+1)+(-1)^n listed in A264080.

List([0..40], n-> Lucas(1,-1,2*n+1)[2] ); # G. C. Greubel, Jul 15 2019

a002878 n = a002878_list !! n
a002878_list = zipWith (+) (tail a001906_list) a001906_list
-- Reinhard Zumkeller, Jan 11 2012

[Lucas(2*n+1): n in [0..40]]; // Vincenzo Librandi, Apr 16 2011

A002878 := proc(n)
    option remember;
    if n <= 1 then
        op(n+1,[1,4]);
    else
        3*procname(n-1)-procname(n-2) ;
    end if;
end proc: # R. J. Mathar, Apr 30 2017

a[n_]:= FullSimplify[GoldenRatio^n - GoldenRatio^-n]; Table[a[n], {n, 1, 40, 2}]
a[1]=1; a[2]=4; a[n_]:=a[n]= 3a[n-1] -a[n-2]; Array[a, 40]
LinearRecurrence[{3, -1}, {1, 4}, 41] (* Jean-François Alcover, Sep 23 2017 *)
Table[Sum[(-1)^Floor[k/2] Binomial[n -Floor[(k+1)/2], Floor[k/2]] 3^(n - k), {k, 0, n}], {n, 0, 40}] (* L. Edson Jeffery, Feb 26 2018 *)
a[ n_] := Fibonacci[2n] + Fibonacci[2n+2]; (* Michael Somos, Jul 31 2018 *)
a[ n_]:= LucasL[2n+1]; (* Michael Somos, Jan 13 2019 *)

a(n)=fibonacci(2*n)+fibonacci(2*n+2) \\ Charles R Greathouse IV, Jun 16 2011

for(n=1,40,q=((1+sqrt(5))/2)^(2*n-1);print1(contfrac(q)[1],", ")) \\ Derek Orr, Jun 18 2015

Vec((1+x)/(1-3*x+x^2) + O(x^40)) \\ Altug Alkan, Oct 26 2015

a002878 = [1, 4]
for n in range(30): a002878.append(3*a002878[-1] - a002878[-2])
print(a002878) # Gennady Eremin, Feb 05 2022

[lucas_number2(2*n+1,1,-1) for n in (0..40)] # G. C. Greubel, Jul 15 2019

a(n+1) = 3*a(n) - a(n-1).
G.f.: (1+x)/(1-3*x+x^2). - Simon Plouffe in his 1992 dissertation
a(n) = S(2*n, sqrt(5)) = S(n, 3) + S(n-1, 3); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, 3) = A001906(n+1) (even-indexed Fibonacci numbers).
a(n) ~ phi^(2*n+1). - Joe Keane (jgk(AT)jgk.org), May 15 2002
Let q(n, x) = Sum_{i=0..n} x^(n-i)*binomial(2*n-i, i); then (-1)^n*q(n, -1) = a(n). - Benoit Cloitre, Nov 10 2002
a(n) = A005248(n+1) - A005248(n) = -1 + Sum_{k=0..n} A005248(k). - Lekraj Beedassy, Dec 31 2002
a(n) = 2^(-n)*A082762(n) = 4^(-n)*Sum_{k>=0} binomial(2*n+1, 2*k)*5^k; see A091042. - Philippe Deléham, Mar 01 2004
a(n) = (-1)^n*Sum_{k=0..n} (-5)^k*binomial(n+k, n-k). - Benoit Cloitre, May 09 2004
From Paul Barry, May 27 2004: (Start)
Both bisection and binomial transform of A000204.
a(n) = Fibonacci(2n) + Fibonacci(2n+2). (End)
Sequence lists the numerators of sinh((2*n-1)*psi) where the denominators are 2; psi=log((1+sqrt(5))/2). Offset 1. a(3)=11. - Al Hakanson (hawkuu(AT)gmail.com), Mar 25 2009
a(n) = A001906(n) + A001906(n+1). - Reinhard Zumkeller, Jan 11 2012
a(n) = floor(phi^(2n+1)), where phi is the golden ratio, A001622. - Thomas Ordowski, Jun 10 2012
a(n) = A014217(2*n+1) = A014217(2*n+2) - A014217(2*n). - Paul Curtz, Jun 11 2013
Sum_{n >= 0} 1/(a(n) + 5/a(n)) = 1/2. Compare with A005248, A001906, A075796. - Peter Bala, Nov 29 2013
a(n) = lim_{m->infinity} Fibonacci(m)^(4n+1)*Fibonacci(m+2*n+1)/ Sum_{k=0..m} Fibonacci(k)^(4n+2). - Yalcin Aktar, Sep 02 2014
From Peter Bala, Mar 22 2015: (Start)
The aerated sequence (b(n))n>=1 = [1, 0, 4, 0, 11, 0, 29, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -1, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy.
b(n) = (1/2)*((-1)^n - 1)*F(n) + (1 + (-1)^(n-1))*F(n+1), where F(n) is a Fibonacci number. The o.g.f. is x*(1 + x^2)/(1 - 3*x^2 + x^4).
Exp( Sum_{n >= 1} 2*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 2*F(n)*x^n.
Exp( Sum_{n >= 1} (-2)*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 2*F(n)*(-x)^n.
Exp( Sum_{n >= 1} 4*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 4*A029907(n)*x^n.
Exp( Sum_{n >= 1} (-4)*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 4*A029907(n)*(-x)^n. Cf. A002315, A004146, A113224 and A192425. (End)
a(n) = sqrt(5*F(2*n+1)^2-4), where F(n) = A000045(n). - Derek Orr, Jun 18 2015
For n > 1, a(n) = 5*F(2*n-1) + L(2*n-3) with F(n) = A000045(n). - J. M. Bergot, Oct 25 2015
For n > 0, a(n) = L(n-1)*L(n+2) + 4*(-1)^n. - J. M. Bergot, Oct 25 2015
For n > 2, a(n) = a(n-2) + F(n+2)^2 + F(n-3)^2 = L(2*n-3) + F(n+2)^2 + F(n-3)^2. - J. M. Bergot, Feb 05 2016 and Feb 07 2016
E.g.f.: ((sqrt(5) - 5)*exp((3-sqrt(5))*x/2) + (5 + sqrt(5))*exp((3+sqrt(5))*x/2))/(2*sqrt(5)). - Ilya Gutkovskiy, Apr 24 2016
a(n) = Sum_{k=0..n} (-1)^floor(k/2)*binomial(n-floor((k+1)/2), floor(k/2))*3^(n-k). - L. Edson Jeffery, Feb 26 2018
a(n)*F(m+2n-1) = F(m+4n-2)-F(m), with Fibonacci number F(m), empirical observation. - Dan Weisz, Jul 30 2018
a(n) = -a(-1-n) for all n in Z. - Michael Somos, Jul 31 2018
Sum_{n>=0} 1/a(n) = A153416. - Amiram Eldar, Nov 11 2020
a(n) = Product_{k=1..n} (1 + 4*sin(2*k*Pi/(2*n+1))^2). - Seiichi Manyama, Apr 30 2021
Sum_{n>=0} (-1)^n/a(n) = (1/sqrt(5)) * A153387 (Carlitz, 1967). - Amiram Eldar, Feb 05 2022
The continued fraction [a(n);a(n),a(n),...] = phi^(2*n+1), with phi = A001622. - A.H.M. Smeets, Feb 25 2022
a(n) = 2*sinh((2*n + 1)*arccsch(2)). - Peter Luschny, May 25 2022
This gives the sequence with 2 1's prepended: b(1)=b(2)=1 and, for k >= 3, b(k) = Sum_{j=1..k-2} (2^(k-j-1) - 1)*b(j). - Neal Gersh Tolunsky, Oct 28 2022 (formula due to Jon E. Schoenfield)
For n > 0, a(n) = 1 + 1/(Sum_{k>=1} F(k)/phi^(2*n*k + k)). - Diego Rattaggi, Nov 08 2023
From Peter Bala, Apr 16 2025: (Start)
a(3*n+1) = a(n)^3 + 3*a(n).
a(5*n+2) = a(n)^5 + 5*a(n)^3 + 5*a(n).
a(7*n+3) = a(n)^7 + 7*a(n)^5 + 14*a(n)^3 + 7*a(n).
For the coefficients see A034807.
The general result is: for k >= 0, a(k*n + (k-1)/2) = 2 * T(k, a(n)/2), where T(k, x) denotes the k-th Chebyshev polynomial of the first kind and a(n) = ((1 + sqrt(5))/2)^(2*n+1) + ((1 - sqrt(5))/2)^(2*n+1).
Sum_{n >= 0} (-1)^n/a(n) = (1/4)* (theta_3(phi) - theta_3(phi^2)) = 0.815947983588122..., where theta_3(x) = 1 + 2*Sum_{n >= 1} x^(n^2) (see A000122) and phi = (sqrt(5) - 1)/2. See Borwein and Borwein, Exercise 3 a, p. 94 and Carlitz, 1967. (End)
From Peter Bala, May 15 2025: (Start)
Sum_{n >= 1} (-1)^(n+1)/(a(n) - 1/a(n)) = 1/5 (telescoping series: 5/(a(n) - 1/a(n)) = 1/A001906(n+1) + 1/A001906(n) ).
More generally, for k >= 1, Sum_{n >= 1} (-1)^(n+1)/(a(k*n) - s(k)/a(k*n)) = 1/(1 + a(k)) where s(k) = a(0) + a(1) + ... + a(k-1) = Lucas(2*k) - 2.
For k >= 1, Sum_{n >= 1} (-1)^(n+1)/(a(n) + L(2*k)^2/a(n)) = (1/5) * A064170(k+2).
Sum_{n >= 1} 1/(a(n) + 9/a(n)) = 3/10 (follows from 1/(a(n) + 9/a(n)) = L(2*n)/A081076(n) - L(2*n+2)/A081076(n+1) ).
More generally, it appears that for k >= 1, Sum_{n >= 1} 1/(a(n) + L(2*k)^2/a(n)) is rational.
Product_{n >= 1} (a(n) + 1)/(a(n) - 1) = sqrt(5) [telescoping product: Product_{k = 1..n} ((a(k) + 1)/(a(k) - 1))^2 = 5*(1 - 4/A240926(n+1)) ]. (End)

Chebyshev and Pell comments from Wolfdieter Lang, Aug 31 2004

A081362 Expansion of q^(1/24) * eta(q) / eta(q^2) in powers of q.

Original entry on oeis.org

1, -1, 0, -1, 1, -1, 1, -1, 2, -2, 2, -2, 3, -3, 3, -4, 5, -5, 5, -6, 7, -8, 8, -9, 11, -12, 12, -14, 16, -17, 18, -20, 23, -25, 26, -29, 33, -35, 37, -41, 46, -49, 52, -57, 63, -68, 72, -78, 87, -93, 98, -107, 117, -125, 133, -144, 157, -168, 178, -192, 209, -223, 236, -255, 276, -294, 312, -335, 361, -385

Offset: 0

Michael Somos, Mar 18 2003

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
(Number of partitions of n into an even number of parts) - (number of partitions of n into an odd number of parts). [Fine]
Number 3 of the 130 identities listed in Slater 1952. - Michael Somos, Aug 20 2015

			G.f. = 1 - x - x^3 + x^4 - x^5 + x^6 - x^7 + 2*x^8 - 2*x^9 + 2*x^10 - 2*x^11 + ...
G.f. = 1/q - q^23 - q^71 + q^95 - q^119 + q^143 - q^167 + 2*q^191 - 2*q^215 + ...

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 425, Corollary 1, Eqs. (37)-(40).
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 38, Eq. (22.14).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Vaclav Kotesovec)
Blecher, Aubrey; Brennan, Charlotte; Knopfmacher, Arnold; Mansour, Toufik Counting corners in partitions. Ramanujan J. 39, No. 1, 201-224 (2016).
J. Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc. (N.S.), 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 3, with k=0. - _N. J. A. Sloane_, Aug 31 2014
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 13.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function p_{e-o}(n).
Lucy Joan Slater, Further Identities of the Rogers-Ramanujan Type, Proc. London Math. Soc., Series 2, vol.s2-54, no.2, pp.147-167, (1952).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000009, A000700, A069910, A069911, A072233.

a[ n_] := SeriesCoefficient[ Product[1 - x^k, {k, 1, n, 2}], {x, 0, n}];
a[ n_] := With[ {t = Log[q] / (2 Pi I)}, SeriesCoefficient[ q^(1/24) DedekindEta[t] / DedekindEta[2 t], {q, 0, n}]];
a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {}, {}, x^2, x], {x, 0, n}]; (* Michael Somos, Jan 02 2015 *)
a[ n_] := SeriesCoefficient[ 1 / QPochhammer[ -x, x], {x, 0, n}]; (* Michael Somos, Jan 02 2015 *)
a[ n_] := SeriesCoefficient[ 1 / Product[ 1 + x^k, {k, n}], {x, 0, n}]; (* Michael Somos, Jan 02 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2], {x, 0, n}]; (* Michael Somos, Jan 02 2015 *)
a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {}, {-1}, x, -1] 2, {x, 0, n}]; (* Michael Somos, May 11 2015 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (m / ( 1 - m)^2 / (16 q))^(-1/24), {q, 0, n}]]; (* Michael Somos, May 11 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) / eta(x^2 + A), n))};

from sage.combinat.partition import number_of_partitions_length
print([sum((-1)^k*number_of_partitions_length(n, k) for k in (0..n)) for n in (0..69)]) # Peter Luschny, Aug 03 2015

G.f.: Product_{k>0} (1 - x^(2*k-1)) = Product_{k>0} 1 / (1 + x^k) = 1 + Sum_{k>0} (-x)^k / (Product_{i=1..k} (1 - x^i)).
a(n) = A027187(n)-A027193(n). [Fine]
This is the convolution inverse of A000009 (partitions into distinct parts) - i.e. the negation of the INVERTi transform of A000009. - Franklin T. Adams-Watters, Jan 06 2006
Expansion of chi(-x) = chi(-x^2) / chi(x) = phi(-x) / f(-x) = phi(-x^2) / f(x) = psi(-x) / f(-x^4) = f(-x) / f(-x^2) = f(-x^2) / psi(x) in powers of x where chi(), psi(), phi(), f() are Ramanujan theta functions.
Expansion of chi(x) * chi(-x) = f(x) / psi(x) = f(-x) / psi(-x) in powers of x^2 where chi(), psi(), f() are Ramanujan theta functions.
Expansion of f(-x, -x^5) / psi(x^3) = phi(-x^3) / f(x, x^2) in powers of x where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Jun 03 2015
Given g.f. A(x), then B(q) = A(q^3)^8 / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2*v + 16*u - v^2.
G.f. A(x) satisfies A(x^2) = A(x) * A(-x).
G.f. is a period 1 Fourier series which satisfies f(-1 / (1152 t)) = 2^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A000009.
Euler transform of period 2 sequence [ -1, 0, ...].
Expansion of q^(1/24) * f1(t) in powers of q = exp(Pi i t) where f1() is a Weber function.
a(n) = (-1)^n * A000700(n). a(2*n) = A069910(n). a(2*n + 1) = - A069911(n).
a(n) ~ (-1)^n * exp(Pi*sqrt(n/6)) / (2*24^(1/4)*n^(3/4)). - Vaclav Kotesovec, Feb 25 2015
a(n) = Sum_{k = 0..n} (-1)^k*A072233(n,k). - Peter Luschny, Aug 03 2015
G.f.: Sum_{k>=0} (-1)^k * x^k^2 / (Product_{i=1..k} (1 - x^(2*i))). - Michael Somos, Aug 20 2015
Sum_{j>=0} a(n-A000217(j)) = 0 if n not in A152749, = (-1)^k if n is in A152749, n=k*(3*k-1). [Merca, Corollary 4.1] - R. J. Mathar, Jun 18 2016
a(n) = -(1/n)*Sum_{k=1..n} A000593(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
G.f.: exp(Sum_{k>=1} (-1)^k*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, May 28 2018
G.f.: (1 - x) * Sum_{k >= 0} (-1)^k * x^(k*(k+2)) / (Product_{i = 1..k} (1 - x^(2*i))) = (1 - x)*(1 - x^3) * Sum_{k >= 0} (-1)^k * x^(k*(k+4)) / (Product_{i = 1..k} (1 - x^(2*i)))= (1 - x)*(1 - x^3)*(1 - x^5) *
Sum_{k >= 0} (-1)^k * x^(k*(k+6)) / (Product_{i = 1..k} (1 - x^(2*i))) = .... - Peter Bala, Jan 15 2021

A005875 Theta series of simple cubic lattice; also number of ways of writing a nonnegative integer n as a sum of 3 squares (zero being allowed).

Original entry on oeis.org

1, 6, 12, 8, 6, 24, 24, 0, 12, 30, 24, 24, 8, 24, 48, 0, 6, 48, 36, 24, 24, 48, 24, 0, 24, 30, 72, 32, 0, 72, 48, 0, 12, 48, 48, 48, 30, 24, 72, 0, 24, 96, 48, 24, 24, 72, 48, 0, 8, 54, 84, 48, 24, 72, 96, 0, 48, 48, 24, 72, 0, 72, 96, 0, 6, 96, 96, 24, 48, 96, 48, 0, 36, 48, 120

Offset: 0

N. J. A. Sloane

Number of ordered triples (i, j, k) of integers such that n = i^2 + j^2 + k^2.
The Madelung Coulomb energy for alternating unit charges in the simple cubic lattice is Sum_{n>=1} (-1)^n*a(n)/sqrt(n) = -A085469. - R. J. Mathar, Apr 29 2006
a(A004215(k))=0 for k=1,2,3,... but no other elements of {a(n)} are zero. - Graeme McRae, Jan 15 2007

			Order and signs are taken into account: a(1) = 6 from 1 = (+-1)^2 + 0^2 + 0^2, a(2) = 12 from 2 = (+-1)^2 + (+-1)^2 + 0^2; a(3) = 8 from 3 = (+-1)^2 + (+-1)^2 + (+-1)^2, etc.
G.f. =  1 + 6*q + 12*q^2 + 8*q^3 + 6*q^4 + 24*q^5 + 24*q^6 + 12*q^8 + 30*q^9 + 24*q^10 + ...

H. Cohen, Number Theory, Vol. 1: Tools and Diophantine Equations, Springer-Verlag, 2007, p. 317.
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 107.
H. Davenport, The Higher Arithmetic. Cambridge Univ. Press, 7th ed., 1999, Chapter V.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 109.
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 54.
L. Kronecker, Crelle, Vol. LVII (1860), p. 248; Werke, Vol. IV, p. 188.
C. J. Moreno and S. S. Wagstaff, Jr., Sums of Squares of Integers, Chapman and Hall, 2006, p. 43.
T. Nagell, Introduction to Number Theory, Wiley, 1951, p. 194.
W. Sierpiński, 1925. Teorja Liczb. pp. 1-410 (p.61).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
H. J. S. Smith, Report on the Theory of Numbers, reprinted in Vol. 1 of his Collected Math. Papers, Chelsea, NY, 1979, see p. 338, Eq. (B').

T. D. Noe, Table of n, a(n) for n = 0..10000
George E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016. See Lemma 2.1.
P. T. Bateman, On the representations of a number as the sum of three squares, Trans. Amer. Math. Soc. 71 (1951), 70-101.
S. Bhargava and C. Adiga, A basic bilateral series summation formula and its applications, Integral Transforms and Special Functions, 2 (1994), 165-184.
J. M. Borwein, K-K S. Choi, On Dirichlet series for sums of squares, Raman. J. 7 (2003) 95-127
S. K. K. Choi, A. V. Kumchev, and R. Osburn, On sums of three squares, arXiv:math/0502007 [math.NT], 2005.
M. Doring, J. Haidenbauer, U.-G. Meissner, and A. Rusetsky, Dynamical coupled-channel approaches on a momentum lattice, arXiv:1108.0676 [hep-lat], 2011.
J. A. Ewell, Recursive determination of the enumerator for sums of three squares, Internat. J. Math. and Math. Sci, 24 (2000), 529-532.
O. Fraser and B. Gordon, On representing a square as the sum of three squares, Amer. Math. Monthly, 76 (1969), 922-923.
M. D. Hirschhorn and J. A. Sellers, On Representations of a Number as a Sum of Three Squares, Discrete Mathematics 199 (1999), 85-101.
M. D. Hirschhorn and J. A. Sellers, On Representations Of A Number As A Sum Of Three Squares
A. Martinez Torres, L. R. Dai, C. Koren, D. Jido, and E. Oset, The KD, eta D_s interaction in finite volume and the D_{s^*0}(2317) resonance, arXiv:1109.0396 [hep-lat], 2011.
R. J. Mathar, Hierarchical Subdivision of the Simple Cubic Lattice, arXiv:1309.3705 [math.MG], 2013.
S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149.
J. L. Mordell, The Representation Of Integers By Three Positive Squares, Michigan Math. J. 7(3): 289-290 (1960).
Eric T. Mortenson, A Kronecker-type identity and the representations of a number as a sum of three squares, arXiv:1702.01627 [math.NT], 2017.
G. Nebe and N. J. A. Sloane, Home page for this lattice
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Theta Series
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to sums of squares

Row d=3 of A122141 and of A319574, 3rd column of A286815.
Cf. A074590 (primitive solutions), A117609 (partial sums), A004215 (positions of zeros).
Analog for 4 squares: A000118.
x^2+y^2+k*z^2: A005875, A014455, A034933, A169783, A169784.
Cf. A000164, A004013, A004015, A008443, A045828, A045831, A045834, A213384.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

# JacobiTheta3 is defined in A000122.
A005875List(len) = JacobiTheta3(len, 3)
A005875List(75) |> println # Peter Luschny, Mar 12 2018

Basis( ModularForms( Gamma1(4), 3/2), 75) [1]; /* Michael Somos, Jun 25 2014 */

(sum(x^(m^2),m=-10..10))^3; seq(coeff(%,x,n), n=0..50);
Alternative:
A005875list := proc(len) series(JacobiTheta3(0, x)^3, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end: A005875list(75); # Peter Luschny, Oct 02 2018

SquaresR[3,Range[0,80]] (* Harvey P. Dale, Jul 21 2011 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^3, {q, 0, n}]; (* Michael Somos, Jun 25 2014 *)
a[ n_] := Length @ FindInstance[ n == x^2 + y^2 + z^2, {x, y, z}, Integers, 10^9]; (* Michael Somos, May 21 2015 *)
QP = QPochhammer; CoefficientList[(QP[q^2]^5/(QP[q]*QP[q^4])^2)^3 + O[q]^80, q] (* Jean-François Alcover, Nov 24 2015 *)

{a(n) = if( n<0, 0, polcoeff( sum( k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n))^3, n))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / (eta(x + A) * eta(x^4 + A))^2)^3, n))}; /* Michael Somos, Jun 03 2012 */

{a(n) = my(G); if( n<0, 0, G = [ 1, 0, 0; 0, 1, 0; 0, 0, 1]; polcoeff( 1 + 2 * x * Ser( qfrep( G, n)), n))}; /* Michael Somos, May 21 2015 */

# uses Python code for A004018
from math import isqrt
def A005875(n): return A004018(n)+(sum(A004018(n-k**2) for k in range(1,isqrt(n)+1))<<1) # Chai Wah Wu, Jun 21 2024

Q = DiagonalQuadraticForm(ZZ, [1]*3)
Q.representation_number_list(75) # Peter Luschny, Jun 20 2014

A number n is representable as the sum of 3 squares iff n is not of the form 4^a (8k+7) (cf. A000378).
There is a classical formula (essentially due to Gauss):
For sums of 3 squares r_3(n): write (uniquely) -n=D(2^vf)^2, with D<0 fundamental discriminant, f odd, v>=-1. Then r_3(n) = 12L((D/.),0)(1-(D/2)) Sum_{d | f} mu(d)(D/d)sigma(f/d).
Here mu is the Moebius function, (D/2) and (D/d) are Kronecker-Legendre symbols, sigma is the sum of divisors function, L((D/.),0)=h(D)/(w(D)/2) is the value at 0 of the L function of the quadratic character (D/.), equal to the class number h(D) divided by 2 or 3 in the special cases D=-4 and -3. - Henri Cohen (Henri.Cohen(AT)math.u-bordeaux1.fr), May 12 2010
a(n) = 3*T(n) if n == 1,2,5,6 mod 8, = 2*T(n) if n == 3 mod 8, = 0 if n == 7 mod 8 and = a(n/4) if n == 0 mod 4, where T(n) = A117726(n). [Moreno-Wagstaff].
"If 12E(n) is the number of representations of n as a sum of three squares, then E(n) = 2F(n) - G(n) where G(n) = number of classes of determinant -n, F(n) = number of uneven classes." - Dickson, quoting Kronecker. [Cf. A117726.]
a(n) = Sum_{d^2|n} b(n/d^2), where b() = A074590() gives the number of primitive solutions.
Expansion of phi(q)^3 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Oct 25 2006.
Euler transform of period 4 sequence [ 6, -9, 6, -3, ...]. - Michael Somos, Oct 25 2006
G.f.: (Sum_{k in Z} x^(k^2))^3.
a(8*n + 7) = 0. a(4*n) = a(n).
a(n) = A004015(2*n) = A014455(2*n) = A004013(4*n) = A169783(4*n). a(4*n + 1) = 6 * A045834(n). a(8*n + 3) = 8 * A008443(n). a(8*n + 5) = 24 * A045831(n). - Michael Somos, Jun 03 2012
a(4*n + 2) = 12 * A045828(n). - Michael Somos, Sep 03 2014
a(n) = (-1)^n * A213384(n). - Michael Somos, May 21 2015
a(n) = (6/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017
a(n) = A004018(n) + 2*Sum_{k=1..floor(sqrt(n))} A004018(n - k^2). - Daniel Suteu, Aug 27 2021
Convolution cube of A000122. Convolution of A004018 and A000122. - R. J. Mathar, Aug 03 2025

More terms from James Sellers, Aug 22 2000

A058696 Number of ways to partition 2n into positive integers.

Original entry on oeis.org

1, 2, 5, 11, 22, 42, 77, 135, 231, 385, 627, 1002, 1575, 2436, 3718, 5604, 8349, 12310, 17977, 26015, 37338, 53174, 75175, 105558, 147273, 204226, 281589, 386155, 526823, 715220, 966467, 1300156, 1741630, 2323520, 3087735, 4087968, 5392783, 7089500, 9289091

Offset: 0

N. J. A. Sloane, Dec 31 2000

nonn

A bisection of A000041, the other one is A058695.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). - Michael Somos, Feb 16 2014
a(n) is the number of partitions of 3n-2 having n as a part, for n >= 1.  Also, a(n+1) is the number of partitions of 3n having n as a part, for n >= 1.  - Clark Kimberling, Mar 02 2014

			G.f. = 1 + 2*x + 5*x^2 + 11*x^3 + 22*x^4 + 42*x^5 + 77*x^6 + 135*x^7 + ...
G.f. = q^-1 + 2*q^47 + 5*q^95 + 11*q^143 + 22*q^191 + 42*q^239 + 77*q^287 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Roland Bacher and Pierre De La Harpe, Conjugacy growth series of some infinitely generated groups, International Mathematics Research Notices, 2016, pp.1-53. (hal-01285685v2)
K. Blum, Bounds on the Number of Graphical Partitions, arXiv:2103.03196 [math.CO], 2021. See Table on p. 7.
Álvaro Gutiérrez and Mercedes H. Rosas, Partial symmetries of iterated plethysms, arXiv:2201.00240 [math.CO], 2022.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000041, A035294, A058695.

a:= n-> combinat[numbpart](2*n):
seq(a(n), n=0..42);  # Alois P. Heinz, Jan 29 2020

nn=100;Table[CoefficientList[Series[Product[1/(1-x^i),{i,1,nn}],{x,0,nn}],x][[2i-1]],{i,1,nn/2}] (* Geoffrey Critzer, Sep 28 2013 *)
(* also *)
Table[PartitionsP[2 n], {n, 0, 40}] (* Clark Kimberling, Mar 02 2014 *)
(* also *)
Table[Count[IntegerPartitions[3 n - 2], p_ /; MemberQ[p, n]], {n, 20}]   (* Clark Kimberling, Mar 02 2014 *)
nmax = 60; CoefficientList[Series[Product[(1 + x^(8*k-4))*(1 + x^(8*k))*(1 + x^k)^2/((1 + x^(8*k-1))*(1 + x^(8*k-7))*(1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 17 2016 *)

combinat::partitions::count(2*i) $i=0..54 // Zerinvary Lajos, Apr 16 2007

{a(n) = if( n<0, 0, polcoeff( 1 / eta(x + O(x^(2*n + 1))), 2*n))}; /* Michael Somos, Apr 25 2003 */

a(n) = numbpart(2*n); \\ Michel Marcus, Sep 28 2013

Expansion of f(x^3, x^5) / f(-x)^2 in powers of x where f() is a Ramanujan theta function. - Michael Somos, Feb 16 2014
Euler transform of period 16 sequence [ 2, 2, 3, 2, 3, 1, 2, 1, 2, 1, 3, 2, 3, 2, 2, 1, ...]. - Michael Somos,  Apr 25 2003
a(n) = A000041(2*n).
Convolution of A000041 and A035294. - Michael Somos, Feb 16 2014
G.f.: Product_{k>=1} (1 + x^(8*k-4)) * (1 + x^(8*k)) * (1 + x^k)^2 / ((1 + x^(8*k-1)) * (1 + x^(8*k-7)) * (1 - x^k)). - Vaclav Kotesovec, Nov 17 2016
a(n) ~ exp(2*Pi*sqrt(n/3)) / (8*sqrt(3)*n). - Vaclav Kotesovec, Feb 16 2022

A067659 Number of partitions of n into distinct parts such that number of parts is odd.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 16, 19, 23, 27, 32, 38, 44, 52, 61, 71, 82, 96, 111, 128, 148, 170, 195, 224, 256, 293, 334, 380, 432, 491, 557, 630, 713, 805, 908, 1024, 1152, 1295, 1455, 1632, 1829, 2048, 2291, 2560, 2859, 3189, 3554, 3958, 4404

Offset: 0

Naohiro Nomoto, Feb 23 2002

Ramanujan theta functions: phi(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

			From _Gus Wiseman_, Jan 09 2021: (Start)
The a(5) = 1 through a(15) = 14 partitions (A-F = 10..15):
  5   6     7     8     9     A     B     C     D     E     F
      321   421   431   432   532   542   543   643   653   654
                  521   531   541   632   642   652   743   753
                        621   631   641   651   742   752   762
                              721   731   732   751   761   843
                                    821   741   832   842   852
                                          831   841   851   861
                                          921   931   932   942
                                                A21   941   951
                                                      A31   A32
                                                      B21   A41
                                                            B31
                                                            C21
                                                            54321
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Joerg Arndt, Matters Computational (The Fxtbook), end of section 16.4.2 "Partitions into distinct parts", pp.348ff
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160 (March 2016), Pages 60-75, function q_o(n).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Dominates A000009.
Numbers with these strict partitions as binary indices are A000069.
The non-strict version is A027193.
The Heinz numbers of these partitions are A030059.
The even version is A067661.
The version for rank is A117193, with non-strict version A101707.
The ordered version is A332304, with non-strict version A166444.
Other cases of odd length:
- A024429 counts set partitions of odd length.
- A089677 counts ordered set partitions of odd length.
- A174726 counts ordered factorizations of odd length.
- A339890 counts factorizations of odd length.
A008289 counts strict partitions by sum and length.
A026804 counts partitions whose least part is odd, with strict case A026832.
Cf. A000700, A027187, A030229, A117192, A332305.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 01 2014

b[n_, i_, t_] := b[n, i, t] = If[n > i*(i + 1)/2, 0, If[n == 0, t, Sum[b[n - i*j, i - 1, Abs[t - j]], {j, 0, Min[n/i, 1]}]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jan 16 2015, after Alois P. Heinz *)
CoefficientList[Normal[Series[(QPochhammer[-x, x]-QPochhammer[x])/2, {x, 0, 100}]], x] (* Andrey Zabolotskiy, Apr 12 2017 *)
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&OddQ[Length[#]]&]],{n,0,30}] (* Gus Wiseman, Jan 09 2021 *)

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)/eta(x+A) - eta(x+A))/2, n))} /* Michael Somos, Feb 14 2006 */

N=66;  q='q+O('q^N);  S=1+2*sqrtint(N);
gf=sum(n=1,S, (n%2!=0) * q^(n*(n+1)/2) / prod(k=1,n, 1-q^k ) );
concat( [0], Vec(gf) )  /* Joerg Arndt, Oct 20 2012 */

N=66;  q='q+O('q^N);  S=1+sqrtint(N);
gf=sum(n=1, S, q^(2*n^2-n) / prod(k=1, 2*n-1, 1-q^k ) );
concat( [0], Vec(gf) )  \\ Joerg Arndt, Apr 01 2014

For g.f. see under A067661.
a(n) = (A000009(n)-A010815(n))/2. - Vladeta Jovovic, Feb 24 2002
Expansion of (1-phi(-q))/(2*chi(-q)) in powers of q where phi(),chi() are Ramanujan theta functions. - Michael Somos, Feb 14 2006
G.f.: sum(n>=1, q^(2*n^2-n) / prod(k=1..2*n-1, 1-q^k ) ). [Joerg Arndt, Apr 01 2014]
a(n) = A067661(n) - A010815(n). - Andrey Zabolotskiy, Apr 12 2017
A000009(n) = a(n) + A067661(n). - Gus Wiseman, Jan 09 2021

A002448 Expansion of Jacobi theta function theta_4(x).

Original entry on oeis.org

1, -2, 0, 0, 2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Number 2 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016

			G.f. = 1 - 2*q + 2*q^4 - 2*q^9 + 2*q^16 - 2*q^25 + 2*q^36 - 2*q^49 + ...

Richard Bellman, A Brief Introduction to Theta Functions, Dover, 2013.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 93, Eq. (34.11), p. 6, Eq. (7.324).
J. Tannery and J. Molk, Eléments de la Théorie des Fonctions Elliptiques, Vol. 2, Gauthier-Villars, Paris, 1902; Chelsea, NY, 1972, see p. 27.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, 4th ed., 1963, p. 464.

T. D. Noe, Table of n, a(n) for n = 0..10000
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 103.
J. W. L. Glaisher, On the deduction of series from infinite products, Messenger of Math., 2 (1873), p. 141.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Eric Weisstein's World of Mathematics, q-Series Identities
D. Zagier, Elliptic modular forms and their applications in "The 1-2-3 of modular forms", Springer-Verlag, 2008

Cf. A000122, A000203, A010054, A089802.

using Nemo
function JacobiTheta4(len, r)
    R, x = PolynomialRing(ZZ, "x")
    e = theta_qexp(r, len, -x)
    [fmpz(coeff(e, j)) for j in 0:len - 1] end
A002448List(len) = JacobiTheta4(len, 1)
A002448List(105) |> println # Peter Luschny, Mar 12 2018

Maple
```
Sum((-x)^(m^2),m=-10..10);
```

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q], {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
QP = QPochhammer; s = QP[q]^2/QP[q^2] + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)

{a(n) = if( n<0, 0, (-1)^n * issquare(n) * 2 - (n==0))}; /* Michael Somos, Jun 17 1999 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 / eta(x^2 + A), n))}; /* Michael Somos, May 01 2003 */

from sympy.ntheory.primetest import is_square
def A002448(n): return (-is_square(n) if n&1 else is_square(n))<<1 if n else 1 # Chai Wah Wu, May 17 2023

Expansion of phi(-q) in powers of q where phi() is a Ramanujan theta function.
Expansion of eta(q)^2 / eta(q^2) in powers of q. - Michael Somos, May 01 2003
Expansion of 2 * sqrt( k' * K / (2 Pi) ) in powers of q. - Michael Somos, Nov 30 2013
Euler transform of period 2 sequence [ -2, -1, ...]. - Michael Somos, May 01 2003
G.f.: Sum_{k in Z} (-1)^k * x^(k^2) = Product_{k>0} (1 - x^k) / (1 + x^k). - Michael Somos, May 01 2003.
G.f.: 1 - 2 Sum_{k>0} x^k/(1 - x^k) Product_{j=1..k} (1 - x^j) / (1 + x^j). - Michael Somos, Apr 12 2012
a(n) = -2 * b(n) where b(n) is multiplicative and b(2^e) = (-1)^(e/2) if e even, b(p^e) = 1 if p>2 and e even, otherwise 0. - Michael Somos, Jul 07 2006
a(3*n + 1) = -2 * A089802(n), a(9*n) = a(n). - Michael Somos, Jul 07 2006
a(3*n + 2) = a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A000122(n). a(n) = (-1)^n * A000122(n). a(8*n + 1) = -2 * A010054(n). - Michael Somos, Apr 12 2012
For n > 0, a(n) = 2*(floor(sqrt(n))-floor(sqrt(n-1)))*(-1)^(floor(sqrt(n))). - Mikael Aaltonen, Jan 17 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 32^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A010054. - Michael Somos, May 05 2016
a(0) = 1, a(n) = -(2/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017
G.f.: exp(Sum_{k>=1} (sigma(k) - sigma(2*k))*x^k/k). - Ilya Gutkovskiy, Sep 19 2018
From Peter Bala, Feb 19 2021: (Start)
G.f: A(q) = eta(q^2)^5 / ( eta(-q)*eta(q^4) )^2.
A(q) = 1 + 2*Sum_{n >= 1} (-1)^n*q^(n*(n+1)/2)/( (1 + q^n) * Product_{k = 1..n} 1 - q^k ).
A(-q)^2 = 1 + 4*Sum_{n >= 1} (-1)^(n+1)*q^(2*n-1)/(1 - q^(2*n-1)), which gives the number of representations of an integer as a sum of two squares. See, for example, Fine, 26.63.
The unsigned sequence has the g.f. 1 + 2*Sum_{n >= 1} q^(n*(n+1)/2) * ( Product_{k = 1..n-1} 1 + q^k ) /( Product_{k = 1..n} 1 + q^(2*k) ) = 1 + 2*q + 2*q^4 + 2*q^9 + .... See Fine, equation 14.43. (End)
Form Peter Bala, Sep 27 2023: (Start)
G.f. A(q) satisfies A(q)*A(-q) = A(q^2)^2.
A(q) = Sum_{n >= 1} (-q)^(n-1)*Product_{k >= n} 1 - q^k. (End)

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A004009 Expansion of Eisenstein series E_4(q) (alternate convention E_2(q)); theta series of E_8 lattice.

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A015128 Number of overpartitions of n: an overpartition of n is an ordered sequence of nonincreasing integers that sum to n, where the first occurrence of each integer may be overlined.

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A027187 Number of partitions of n into an even number of parts.

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A004018 Theta series of square lattice (or number of ways of writing n as a sum of 2 squares). Often denoted by r(n) or r_2(n).

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A002878 Bisection of Lucas sequence: a(n) = L(2*n+1).

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