A274621 -id:A274621 - cp's OEIS frontend

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.

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

A008441 Number of ways of writing n as the sum of 2 triangular numbers.

Original entry on oeis.org

1, 2, 1, 2, 2, 0, 3, 2, 0, 2, 2, 2, 1, 2, 0, 2, 4, 0, 2, 0, 1, 4, 2, 0, 2, 2, 0, 2, 2, 2, 1, 4, 0, 0, 2, 0, 4, 2, 2, 2, 0, 0, 3, 2, 0, 2, 4, 0, 2, 2, 0, 4, 0, 0, 0, 4, 3, 2, 2, 0, 2, 2, 0, 0, 2, 2, 4, 2, 0, 2, 2, 0, 3, 2, 0, 0, 4, 0, 2, 2, 0, 6, 0, 2, 2, 0, 0, 2, 2, 0, 1, 4, 2, 2, 4, 0, 0, 2, 0, 2, 2, 2, 2, 0, 0

Offset: 0

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). The present sequence gives the expansion coefficients of psi(q)^2.

Also the number of positive odd solutions to equation x^2 + y^2 = 8*n + 2. - Seiichi Manyama, May 28 2017

			G.f. = 1 + 2*x + x^2 + 2*x^3 + 2*x^4 + 3*x^6 + 2*x^7 + 2*x^9 + 2*x^10 + 2*x^11 + ...
G.f. for B(q) = q * A(q^4) = q + 2*q^5 + q^9 + 2*q^13 + 2*q^17 + 3*q^25 + 2*q^29 + 2*q^37 + 2*q^41 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag. See p. 139 Example (iv).
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
R. W. Gosper, Strip Mining in the Abandoned Orefields of Nineteenth Century Mathematics, in Computers in Mathematics (Ed. D. V. Chudnovsky and R. D. Jenks). New York: Dekker, 1990. See p. 279.
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 Pi_q.]
P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916. See vol. 2, p 31, Article 272.
Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, John Wiley and Sons, Inc., NY 1991, p. 165.

T. D. Noe, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 575, 16.23.1 and 16.23.2.
R. P. Agarwal, Lambert series and Ramanujan, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293 (see p. 285).
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 72, Eq (31.2); p. 78, Eq. following (32.25).
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167. See p. 108.
R. W. Gosper, Experiments and discoveries in q-trigonometry, Preprint.
R. W. Gosper, q-Trigonometry: Some Prefatory Afterthoughts.
Michael D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
Nadia 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.
Nicco, Conjectured identity of the product of two theta functions, Math Stackexchange, Sep 09 2015.
K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp. 73-94.
Hjalmar Rosengren, Sums of triangular numbers from the Frobenius determinant, arXiv:math/0504272 [math.NT], 2005.
James A. Sellers and Roberto Tauraso, Arithmetic properties of MacMahon-type sums of divisors: the odd case, arXiv:2505.12813 [math.NT], 2025.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

A002654, A008442, A035154, A035181, A035184, A112301, A113406, A113414, A113446, A113652 all satisfy A(4n+1)=a(n).
Cf. A004020, A005883, A104794, A052343, A199015 (partial sums).
Number of ways of writing n as a sum of k triangular numbers, for k=1,...: A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787, A014809.
Cf. A002175, A053692, A113407, A121444.
Cf. A000217, A001938, A004018.
Cf. A274621 (reciprocal series).
Cf. A001826, A001842.
Cf. A019669, A079006.

a052343 = (flip div 2) . (+ 1) . a008441
-- Reinhard Zumkeller, Jul 25 2014

A := Basis( ModularForms( Gamma1(8), 1), 420); A[2]; /* Michael Somos, Jan 31 2015 */

sigmamr := proc(n,m,r) local a,d ; a := 0 ; for d in numtheory[divisors](n) do if modp(d,m) = r then a := a+1 ; end if; end do: a; end proc:
A002654 := proc(n) sigmamr(n,4,1)-sigmamr(n,4,3) ; end proc:
A008441 := proc(n) A002654(4*n+1) ; end proc:
seq(A008441(n),n=0..90) ; # R. J. Mathar, Mar 23 2011

Plus@@((-1)^(1/2 (Divisors[4#+1]-1)))& /@ Range[0, 104] (* Ant King, Dec 02 2010 *)
a[ n_] := SeriesCoefficient[ (1/2) EllipticTheta[ 2, 0, q] EllipticTheta[ 3, 0, q], {q, 0, n + 1/4}]; (* Michael Somos, Jun 19 2012 *)
a[ n_] := SeriesCoefficient[ (1/4) EllipticTheta[ 2, 0, q]^2, {q, 0, 2 n + 1/2}]; (* Michael Somos, Jun 19 2012 *)
a[ n_] := If[ n < 0, 0, DivisorSum[ 4 n + 1, (-1)^Quotient[#, 2] &]];  (* Michael Somos, Jun 08 2014 *)
QP = QPochhammer; s = QP[q^2]^4/QP[q]^2 + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)
TriangleQ[n_] := IntegerQ@Sqrt[8n +1]; Table[Count[FrobeniusSolve[{1, 1}, n], {?TriangleQ}], {n, 0, 104}] (* Robert G. Wilson v, Apr 15 2017 *)

{a(n) = if( n<1, n==0, polcoeff( sum(k=0, (sqrtint(8*n + 1) - 1)\2, x^(k * (k+1)/2), x * O(x^n))^2, n) )};

{a(n) = if( n<0, 0, n = 4*n + 1; sumdiv(n, d, (-1)^(d\2)))}; /* Michael Somos, Sep 02 2005 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 / eta(x + A)^2, n))};

{a(n) = if( n<0, 0, n = 4*n + 1; sumdiv( n, d, (d%4==1) - (d%4==3)))}; /* Michael Somos, Sep 14 2005 */

{ my(q='q+O('q^166)); Vec(eta(q^2)^4 / eta(q)^2) } \\ Joerg Arndt, Apr 16 2017

ModularForms( Gamma1(8), 1, prec=420).1; # Michael Somos, Jun 08 2014

This sequence is the quadrisection of many sequences. Here are two examples:
a(n) = A002654(4n+1), the difference between the number of divisors of 4*n+1 of form 4*k+1 and the number of form 4*k-1. - David Broadhurst, Oct 20 2002
a(n) = b(4*n + 1), where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = e+1 if p == 1 (mod 4). - Michael Somos, Sep 14 2005
G.f.: (Sum_{k>=0} x^((k^2 + k)/2))^2 = (Sum_{k>=0} x^(k^2 + k)) * (Sum_{k in Z} x^(k^2)).
Expansion of Jacobi theta (theta_2(0, sqrt(q)))^2 / (4 * q^(1/4)).
Sum[d|(4n+1), (-1)^((d-1)/2) ].
Given g.f. A(x), then B(q) = q * A(q^4) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v^3 + 4 * v * w^2 - u^2 * w. - Michael Somos, Sep 14 2005
Given g.f. A(x), then B(q) = q * A(q^4) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1 * u3 - (u2 - u6) * (u2 + 3*u6). - Michael Somos, Sep 14 2005
Expansion of Jacobi k/(4*q^(1/2)) * (2/Pi)* K(k) in powers of q^2. - Michael Somos, Sep 14 2005. Convolution of A001938 and A004018. This appears in the denominator of the Jacobi sn and cn formula given in the Abramowitz-Stegun reference, p. 575, 16.23.1 and 16.23.2, where m=k^2. - Wolfdieter Lang, Jul 05 2016
G.f.: Sum_{k>=0} a(k) * x^(2*k) = Sum_{k>=0} x^k / (1 + x^(2*k + 1)).
G.f.: Sum_{k in Z} x^k / (1 - x^(4*k + 1)). - Michael Somos, Nov 03 2005
Expansion of psi(x)^2 = phi(x) * psi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.
Moebius transform is period 8 sequence [ 1, -1, -1, 0, 1, 1, -1, 0, ...]. - Michael Somos, Jan 25 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 1/2 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A104794.
Euler transform of period 2 sequence [ 2, -2, ...].
G.f.: q^(-1/4) * eta(q^2)^4 / eta(q)^2. See also the Fine reference.
a(n) = Sum_{k=0..n} A010054(k)*A010054(n-k). - Reinhard Zumkeller, Nov 03 2009
A004020(n) = 2 * a(n). A005883(n) = 4 * a(n).
Convolution square of A010054.
G.f.: Product_{k>0} (1 - x^(2*k))^2 / (1 - x^(2*k-1))^2.
a(2*n) = A113407(n). a(2*n + 1) = A053692(n). a(3*n) = A002175(n). a(3*n + 1) = 2 * A121444(n). a(9*n + 2) = a(n). a(9*n + 5) = a(9*n + 8) = 0. - Michael Somos, Jun 08 2014
G.f.: exp( Sum_{n>=1} 2*(x^n/n) / (1 + x^n) ). - Paul D. Hanna, Mar 01 2016
a(n) = A001826(2+8*n) - A001842(2+8*n), the difference between the number of divisors 1 (mod 4) and 3 (mod 4) of 2+8*n. See the Ono et al. link, Corollary 1, or directly the Niven et al. reference, p. 165, Corollary (3.23). - Wolfdieter Lang, Jan 11 2017
Expansion of continued fraction 1 / (1 - x^1 + x^1*(1 - x^1)^2 / (1 - x^3 + x^2*(1 - x^2)^2 / (1 - x^5 + x^3*(1 - x^3)^2 / ...))) in powers of x^2. - Michael Somos, Apr 20 2017
Given g.f. A(x), and B(x) is the g.f. for A079006, then B(x) = A(x^2) / A(x) and B(x) * B(x^2) * B(x^4) * ... = 1 / A(x). - Michael Somos, Apr 20 2017
a(0) = 1, a(n) = (2/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017
From Paul D. Hanna, Aug 10 2019: (Start)
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (x^(2*n+1) - x^(2*k))^(n-k) = Sum_{n>=0} a(n)*x^(2*n).
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (x^(2*n+1) + x^(2*k))^(n-k) * (-1)^k = Sum_{n>=0} a(n)*x^(2*n). (End)
From Peter Bala, Jan 05 2021: (Start)
G.f.: Sum_{n = -oo..oo} x^(4*n^2+2*n) * (1 + x^(4*n+1))/(1 - x^(4*n+1)). See Agarwal, p. 285, equation 6.20 with i = j = 1 and mu = 4.
For prime p of the form 4*k + 3, a(n*p^2 + (p^2 - 1)/4) = a(n).
If n > 0 and p are coprime then a(n*p + (p^2 - 1)/4) = 0. The proofs are similar to those given for the corresponding results for A115110. Cf. A000729.
For prime p of the form 4*k + 1 and for n not congruent to (p - 1)/4 (mod p) we have a(n*p^2 + (p^2 - 1)/4) = 3*a(n) (since b(n), where b(4*n+1) = a(n), is multiplicative). (End)
From Peter Bala, Mar 22 2021: (Start)
G.f. A(q) satisfies:
A(q^2) = Sum_{n = -oo..oo} q^n/(1 - q^(4*n+2)) (set z = q, alpha = q^2, mu = 4 in Agarwal, equation 6.15).
A(q^2) = Sum_{n = -oo..oo} q^(2*n)/(1 - q^(4*n+1)) (set z = q^2, alpha = q, mu = 4 in Agarwal, equation 6.15).
A(q^2) = Sum_{n = -oo..oo} q^n/(1 + q^(2*n+1))^2 = Sum_{n = -oo..oo} q^(3*n+1)/(1 + q^(2*n+1))^2. (End)
G.f.: Sum_{k>=0} a(k) * q^k = Sum_{k>=0} (-1)^k * q^(k*(k+1)) + 2 * Sum_{n>=1, k>=0} (-1)^k * q^(k*(k+2*n+1)+n). - Mamuka Jibladze, May 17 2021
G.f.: Sum_{k>=0} a(k) * q^k = Sum_{k>=0} (-1)^k * q^(k*(k+1)) * (1 + q^(2*k+1))/(1 - q^(2*k+1)). - Mamuka Jibladze, Jun 06 2021
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 (A019669). - Amiram Eldar, Oct 15 2022

More terms and information from Michael Somos, Mar 23 2003

A366730 Expansion of g.f. A(x,y) satisfying 0 = Sum_{n=-oo..+oo} x^n * A(x,y)^n * (y - x^(n-1))^(n+1), as a triangle of coefficients T(n,k) of x^n*y^k in A(x,y), read by rows n >= 0.

Original entry on oeis.org

1, 0, 1, 0, -2, 2, 0, 3, -6, 5, 0, -6, 14, -20, 14, 0, 11, -36, 59, -70, 42, 0, -18, 87, -176, 246, -252, 132, 0, 28, -190, 500, -824, 1022, -924, 429, 0, -44, 386, -1312, 2615, -3780, 4236, -3432, 1430, 0, 69, -756, 3218, -7734, 13107, -17112, 17523, -12870, 4862, 0, -104, 1443, -7514, 21496, -42444, 64031, -76692, 72358, -48620, 16796

Offset: 0

Paul D. Hanna, Oct 29 2023

			G.f.: A(x,y) = 1 + x*y + x^2*(-2*y + 2*y^2) + x^3*(3*y - 6*y^2 + 5*y^3) + x^4*(-6*y + 14*y^2 - 20*y^3 + 14*y^4) + x^5*(11*y - 36*y^2 + 59*y^3 - 70*y^4 + 42*y^5) + x^6*(-18*y + 87*y^2 - 176*y^3 + 246*y^4 - 252*y^5 + 132*y^6) + x^7*(28*y - 190*y^2 + 500*y^3 - 824*y^4 + 1022*y^5 - 924*y^6 + 429*y^7) + x^8*(-44*y + 386*y^2 - 1312*y^3 + 2615*y^4 - 3780*y^5 + 4236*y^6 - 3432*y^7 + 1430*y^8) + x^9*(69*y - 756*y^2 + 3218*y^3 - 7734*y^4 + 13107*y^5 - 17112*y^6 + 17523*y^7 - 12870*y^8 + 4862*y^9) + ...
where A = A(x,y) satisfies
0 = Sum_{n=-oo..+oo} x^n * A^n * (y - x^(n-1))^(n+1);
explicitly,
0 = ((-A + 1)/A)/x + y + (A*y^2 - 2*A*y + ((A^3 - 1)/A^2))*x + A^2*y^3*x^2 + (A^3*y^4 - 3*A^2*y^2)*x^3 + (A^4*y^5 + ((3*A^4 - 1)/A^2)*y)*x^4 + (A^5*y^6 - 4*A^3*y^3 + ((-A^5 + 1)/A^3))*x^5 + A^6*y^7*x^6 + (A^7*y^8 - 5*A^4*y^4 + ((6*A^5 - 1)/A^2)*y^2)*x^7 + A^8*y^9*x^8 + (A^9*y^10 - 6*A^5*y^5 + ((-4*A^6 + 2)/A^3)*y)*x^9 + (A^10*y^11 + ((10*A^6 - 1)/A^2)*y^3)*x^10 + ...
This triangle of coefficients of x^n*y^k in A(x,y) begins:
1;
0, 1;
0, -2, 2;
0, 3, -6, 5;
0, -6, 14, -20, 14;
0, 11, -36, 59, -70, 42;
0, -18, 87, -176, 246, -252, 132;
0, 28, -190, 500, -824, 1022, -924, 429;
0, -44, 386, -1312, 2615, -3780, 4236, -3432, 1430;
0, 69, -756, 3218, -7734, 13107, -17112, 17523, -12870, 4862;
0, -104, 1443, -7514, 21496, -42444, 64031, -76692, 72358, -48620, 16796;
0, 152, -2668, 16862, -56856, 129425, -223458, 307189, -340912, 298298, -184756, 58786;
0, -222, 4782, -36456, 144159, -375618, 734310, -1143924, 1453221, -1504932, 1227876, -705432, 208012; ...
in which the main diagonal equals the Catalan numbers (A000108), and column 1 equals the coefficients in Product_{n>=1} (1 - q^(2*n-1))^2/(1 - q^(2*n))^2 (A274621).

Paul D. Hanna, Table of n, a(n) for n = 0..1274

Cf. A274621 (column 1), A000108 (diagonal), A366736 (central terms).

Cf. A366731 (y=1), A366732 (y=2), A366733 (y=3), A366734 (y=4), A366735 (y=-1).

{T(n,k) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(n=-#A,#A, x^n * Ser(A)^n * (y - x^(n-1))^(n+1) ), #A-2)); polcoeff(A[n+1],k)}
for(n=0,12, for(k=0,n, print1(T(n,k),", "));print(""))

G.f. A(x,y) = Sum_{n>=0} sum_{k=0..n} T(n,k)*x^n*y^k satisfies the following formulas.
(1) 0 = Sum_{n=-oo..+oo} x^n * A(x,y)^n * (y - x^(n-1))^(n+1).
(2) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( A(x,y)^n * (1 - y*x^(n+1))^(n-1) ).

A273225 Number of bipartitions of n wherein odd parts are distinct (and even parts are unrestricted).

Original entry on oeis.org

1, 2, 3, 6, 11, 18, 28, 44, 69, 104, 152, 222, 323, 460, 645, 902, 1254, 1722, 2343, 3174, 4278, 5722, 7601, 10056, 13250, 17358, 22623, 29382, 38021, 48984, 62857, 80404, 102528, 130282, 165002, 208398, 262495, 329666, 412878, 515840

Offset: 0

M.S. Mahadeva Naika, May 18 2016

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number of bipartitions of 'n' wherein odd parts are distinct (and even parts are unrestricted).
G.f. is the square of the g.f. of A006950. - Vaclav Kotesovec, Mar 25 2017

			a(4)=11 because "(0,4)=(0,3+1)=(0,2+2)=(1,3)=(1,2+1)=(2,2)=(4,0)=(3+1,0)=(2+2,0)=(3,1)=(2+1,1)".
G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 11*x^4 + 18*x^5 + 28*x^6 + 44*x^7 + ... - _Michael Somos_, Mar 02 2019
G.f. = q^-1 + 2*q^3 + 3*q^7 + 6*q^11 + 11*q^15 + 18*q^19 + 28*q^23 + ... - _Michael Somos_, Mar 02 2019

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
M. D. Hirschhorn and J. A. Sellers, Arithmetic properties of partitions with odd parts distinct, Ramanujan J. 22 (2010), 273--284.
M. S. Mahadeva Naika and D. S. Gireesh, Arithmetic Properties of Partition k-tuples with Odd Parts Distinct, JIS, Vol. 19 (2016), Article 16.5.7
Michael Somos, Introduction to Ramanujan theta functions
L. Wang, Arithmetic properties of partition triples with odd parts distinct, Int. J. Number Theory, 11 (2015), 1791--1805.
L. Wang, Arithmetic properties of partition quadruples with odd parts distinct, Bull. Aust. Math. Soc., doi:10.1017/S0004972715000647.
L. Wang, New congruences for partitions where the odd parts are distinct, J. Integer Seq. (2015), article 15.4.2.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

For a version with signs see A274621.

Cf. A006950.

Digits:=200:with(PolynomialTools): with(qseries): with(ListTools):
GenFun:=series(etaq(q,2,100)^2/etaq(q,1,100)^2/etaq(q,4,100)^2,q,50):
CoefficientList(sort(convert(GenFun,polynom),q,ascending),q);

s = QPochhammer[-1, x]^2/(4*QPochhammer[x^4, x^4]^2) + O[x]^40; CoefficientList[s, x] (* Jean-François Alcover, May 20 2016 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2, x^4] / QPochhammer[ x])^2, {x, 0, n}]; (* Michael Somos, Mar 02 2019 *)

{a(n) = my(A); if( n<0, 0 , A = x * O(x^n); polcoeff( eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A))^2, n))}; /* Michael Somos, Mar 02 2019 */

G.f.: Product_{k>=1} (1 + x^k)^2 / (1 - x^(4*k))^2, corrected by Vaclav Kotesovec, Mar 25 2017
Expansion of 1 / psi(-x)^2 in powers of x where psi() is a Ramanujan theta function.
a(n) ~ exp(Pi*sqrt(n))/(2^(5/2)*n^(5/4)). - Vaclav Kotesovec, Jul 05 2016
Euler transform of period 4 sequence [2, 0, 2, 2, ...]. - Michael Somos, Mar 02 2019

A274659 Triangle entry T(n, m) gives the m-th contribution T(n, m)sin((2m+1)v) to the coefficient of q^n in the Fourier expansion of Jacobi's elliptic sn(u|k) function when expressed in the variables v = u/(2K(k)/Pi) and q, the Jacobi nome, written as series in (k/4)^2. K is the real quarter period of elliptic functions.

Original entry on oeis.org

1, 1, 1, -1, 0, 1, -1, -2, 0, 1, 2, 1, -2, 0, 1, 2, 3, 0, -2, 0, 1, -4, -2, 3, 0, -2, 0, 1, -4, -5, 1, 3, 0, -2, 0, 1, 7, 3, -6, 0, 3, 0, -2, 0, 1, 7, 9, -2, -6, 0, 3, 0, -2, 0, 1, -11, -5, 11, 1, -6, 0, 3, 0, -2, 0, 1, -11, -15, 3, 11, 0, -6, 0, 3, 0, -2, 0, 1, 17, 9, -17, -2, 11, 0, -6, 0, 3, 0, -2, 0, 1

Offset: 0

Wolfdieter Lang, Jul 18 2016

If one takes the row polynomials as R(n, x) = Sum_{m=0..n} T(n, m)*x^(2*m+1), n >= 0, Jacobi's elliptic sn(u|k) function in terms of the new variables v and q becomes sn(u|k) = Sum_{n>=0} R(n, x)*q^n, if one replaces in R(n, x) x^j by sin(j*v).
v=v(u,k^2) and q=q(k^2) are computed with the help of A038534/A056982 for (2/Pi)*K and A002103 for q expanded in powers of (k/4)^2.
A test for sn(u|k) with u = 1, k = sqrt(1/2), that is v approximately 0.8472130848 and q approximately 0.04321389673, with rows n=0..10 (q powers not exceeding 10) gives 0.8030018002 to be compared with sn(1|sqrt(1/2)) approximately 0.8030018249.
For the derivation of the Fourier series formula of sn given in Abramowitz-Stegun (but there the notation sn(u|m=k^2) is used for sn(u|k)) see, e.g., Whittaker and Watson, p. 511 or Armitage and Eberlein, Exercises on p. 55.
For the cn expansion see A274661.
See also the W. Lang link, equations (34) and (35).

			The triangle T(n, m) begins:
      m  0   1  2  3  4  5  6  7  8  9 10 11
n\ 2m+1  1   3  5  7  9 11 13 15 17 19 21 23
0:       1
1:       1   1
2:      -1   0  1
3:      -1  -2  0  1
4:       2   1 -2  0  1
5:       2   3  0 -2  0  1
6:      -4  -2  3  0 -2  0  1
7:      -4  -5  1  3  0 -2  0  1
8:       7   3 -6  0  3  0 -2  0  1
9:       7   9 -2 -6  0  3  0 -2  0  1
10:    -11  -5 11  1 -6  0  3  0 -2  0  1
11:    -11 -15  3 11  0 -6  0  3  0 -2  0  1
...
T(4, 0) = 2 from the x^1 term in b(0, x)*a(4) + b(2, x)*a(2) + b(4, x)*a(0), that is x^1*3 + x^1*(-2) + x^1*1 = +2*x^1.
n=4: R(4, x) = 2*x^1 + 1*x^3 - 2*x^5 + 0*x^7 + 1*x^9, that is the sn(u|k) contribution of order q^4 in the new variables v and q is (2*sin(1*v) + 1*sin(3*v) - 2*sin(5*v) + 1*sin(9*v))*q^4.

J. V. Armitage and W. F. Eberlein, Elliptic Functions, London Mathematical Society, Student Texts 67, Cambridge University Press, 2006.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, fourth edition, reprinted, 1958, Cambridge at the University Press.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 375, 16.23.1.
Wolfdieter Lang, Expansions for phase space coordinates for the plane pendulum

Cf. A002103, A038534/A056982, A099774, A274621, A274658, A274661.

T(n, m) = [x^(2*m+1)]Sum_{j=0..n} b(j, x)*a(n-j), with a(k) = A274621(k/2) if k is even and a(k) = 0 if k is odd, and b(j, x) = Sum_{r | 2*j+1} x^r = Sum_{k=1..A099774(j+1)} x^(A274658(j, k)), for j >= 0.

A295832 Expansion of Product_{k>=1} ((1 + x^(2k-1))/(1 - x^(2k)))^k.

Original entry on oeis.org

1, 1, 1, 3, 5, 8, 12, 20, 33, 50, 74, 114, 175, 257, 375, 555, 814, 1171, 1677, 2406, 3435, 4855, 6825, 9591, 13428, 18667, 25851, 35745, 49250, 67544, 92340, 125966, 171345, 232257, 313945, 423470, 569778, 764465, 1023231, 1366827, 1821756, 2422394, 3214318, 4257088, 5627086, 7422941

Offset: 0

Ilya Gutkovskiy, Nov 28 2017

nonn

Cf. A000219, A006950, A113415, A156616, A224364, A263140, A273225, A273226, A274621, A295831.

nmax = 45; CoefficientList[Series[Product[((1 + x^(2 k - 1))/(1 - x^(2 k)))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[Exp[Sum[x^k ((-1)^(k + 1) + x^k)/(k (1 - x^(2 k))^2), {k, 1, nmax}]], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 + x^(2*k-1))/(1 - x^(2*k)))^k.
G.f.: exp(Sum_{k>=1} x^k*((-1)^(k+1) + x^k)/(k*(1 - x^(2*k))^2)).
a(n) ~ exp(3 * (7*Zeta(3))^(1/3) * n^(2/3) / 4 + Pi^2 * n^(1/3) / (24 * (7*Zeta(3))^(1/3)) - Pi^4 / (12096 * Zeta(3)) + 1/12) * (7*Zeta(3))^(7/36) / (A * 2^(23/24) * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 28 2017

A274622 Irregular triangular array read by rows: coefficients in expansion of Gosper's q-sine function sin_q(Pi*z).

Original entry on oeis.org

1, 1, 2, 2, 1, 4, 4, 1, 2, 8, 8, 2, 4, 14, 14, 4, 8, 24, 24, 8, 1, 14, 40, 40, 14, 1, 2, 24, 64, 64, 24, 2, 4, 40, 100, 100, 40, 4, 8, 64, 154, 154, 64, 8, 14, 100, 232, 232, 100, 14, 24, 154, 344, 344, 154, 24, 1, 40, 232, 504, 504, 232, 40, 1, 2, 64, 344, 728, 728, 344, 64, 2, 4, 100, 504, 1040, 1040, 504, 100, 4

Offset: 0

N. J. A. Sloane, Jul 04 2016

			The array begins:
.........1,1,
.........2,2,
.......1,4,4,1,
.......2,8,8,2,
.......4,14,14,4,
.......8,24,24,8,
....1,14,40,40,14,1,
....2,24,64,64,24,2,
....4,40,100,100,40,4,
....8,64,154,154,64,8,
....14,100,232,232,100,14,
....24,154,344,344,154,24,
..1,40,232,504,504,232,40,1,
..2,64,344,728,728,344,64,2,
..4,100,504,1040,1040,504,100,4,
...

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.

R. W. Gosper, Experiments and discoveries in q-trigonometry, Preprint.
R. W. Gosper, q-Trigonometry: Some Prefatory Afterthoughts

Cf. A274623, A274624, A274625.

nmax = 14; kmax = 4; QP = QPochhammer; s = QP[q^2]/QP[q]^2 + O[q]^(nmax + 1); col[1] = CoefficientList [s, q]; col[k_] := Join[Array[0&, k(k-1)], Take[col[1], nmax-k(k-1)+1]]; T = Transpose[Array[col, kmax]]; ro[n_] := DeleteCases[T[[n+1]], 0]; row[n_] := Join[Reverse[ro[n]], ro[n]]; Table[row[n], {n, 0, nmax}] // Flatten (* Jean-François Alcover, Oct 07 2016 *)

A274624 Irregular triangular array read by rows: coefficients in expansion of Gosper's q-cosine function cos_q(Pi*z).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 2, 4, 8, 4, 1, 8, 14, 8, 1, 2, 14, 24, 14, 2, 4, 24, 40, 24, 4, 8, 40, 64, 40, 8, 14, 64, 100, 64, 14, 1, 24, 100, 154, 100, 24, 1, 2, 40, 154, 232, 154, 40, 2, 4, 64, 232, 344, 232, 64, 4, 8, 100, 344, 504, 344, 100, 8, 14, 154, 504, 728, 504, 154, 14, 24, 232, 728, 1040, 728, 232, 24

Offset: 0

N. J. A. Sloane, Jul 05 2016

			The array begins:
.........1,
.......1,2,1,
.......2,4,2,
.......4,8,4,
.....1,8,14,8,1,
....2,14,24,14,2,
....4,24,40,24,4,
....8,40,64,40,8,
...14,64,100,64,14,
.1,24,100,154,100,24,1,
.2,40,154,232,154,40,2,
.4,64,232,344,232,64,4,
.8,100,344,504,344,100,8,
14,154,504,728,504,154,14,
24,232,728,1040,728,232,24,
...

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.

R. W. Gosper, Experiments and discoveries in q-trigonometry, Preprint.
R. W. Gosper, q-Trigonometry: Some Prefatory Afterthoughts

Cf. A274622, A274623, A274625.

A274661 Triangle read by rows: T(n, m) gives the m-th contribution T(n, m)cos((2m+1)v) to the coefficient of q^n in the Fourier expansion of Jacobi's elliptic cn(u|k) function when expressed in the variables v = u/(2K(k)/Pi) and q, the Jacobi nome, written as series in (k/4)^2. K is the real quarter period of elliptic functions.

Original entry on oeis.org

1, -1, 1, -1, 0, 1, 1, -2, 0, 1, 2, -1, -2, 0, 1, -2, 3, 0, -2, 0, 1, -4, 2, 3, 0, -2, 0, 1, 4, -5, -1, 3, 0, -2, 0, 1, 7, -3, -6, 0, 3, 0, -2, 0, 1, -7, 9, 2, -6, 0, 3, 0, -2, 0, 1, -11, 5, 11, -1, -6, 0, 3, 0, -2, 0, 1, 11, -15, -3, 11, 0, -6, 0, 3, 0, -2, 0, 1, 17, -9, -17, 2, 11, 0, -6, 0, 3, 0, -2, 0, 1, -17, 23, 6, -18, -1, 11, 0, -6, 0, 3, 0, -2, 0, 1

Offset: 0

Wolfdieter Lang, Jul 27 2016

If one takes the row polynomials as P(n, x) = Sum_{m=0..n} T(n, m)*x^m, n >= 0, Jacobi's elliptic function cn(u|k) in terms of the new variables v and q becomes cn(u|k) = Sum_{n>=0} P(n, x)*q^n, if in P(n, x) one replaces x^j by cos((2*j+1)*v).
v=v(u,k^2) and q=q(k^2) are computed with the help of A038534/A056982 for (2/Pi)*K and A002103 for q expanded in powers of (k/4)^2.
A test for cn(u|k) with u = 1, k = sqrt(1/2), that is v approximately 0.8472130848 and q approximately 0.04321389673, with rows n=0..10 (q powers not exceeding 10) gives 0.5959766014 to be compared with cn(1|sqrt(1/2)) approximately 0.5959765676.
For the derivation of the Fourier series formula of cn given in Abramowitz-Stegun (but there the notation sn(u|m=k^2) is used for sn(u|k)) see, e.g., Whittaker and Watson, p. 511 or Armitage and Eberlein, Exercises on p. 55.
For sn see A274659 (differently signed triangle).
The sum of entries in row n is P(n, 1) = A000007(n): 1, repeat 0. Proof: due to the g.f. identity (from the convolution)
  Sum_{n >= 0} x^n/(1 + x^(2*n+1))  = (Sum_{n >= 0} x^(n*(n+1)))^2.
  This is proved by bisecting the g.f. on the l.h.s. which generates c(n, 1) = (-1)^n*Sum_{2*r+1 | 2*n+1} (-1)^n. The part with n = 2*k+1 vanishes due to r_2(4*k+1)/4 = 0, where r_2(n) is the number of solutions of n as a sum of two squares. See the Grosswald reference. The part with n = 2*k becomes Sum_{k >= 0} x^(2*k) r_2(4*k+1)/4 which is the r.h.s. See A008441, the Broadhurst Oct 20 2002 comment.
For another version of this expansion of cn see A275791.
See also the W. Lang link, eqs. (43) and (44). - Wolfdieter Lang, Aug 26 2016

			The triangle T(n, m) begins:
      m  0   1  2  3  4  5  6  7  8  9 10 11
n\ 2m+1  1   3  5  7  9 11 13 15 17 19 21 23
0:       1
1:      -1   1
2:      -1   0  1
3:       1  -2  0  1
4:       2  -1 -2  0  1
5:      -2   3  0 -2  0  1
6:      -4   2  3  0 -2  0  1
7:       4  -5 -1  3  0 -2  0  1
8:       7  -3 -6  0  3  0 -2  0  1
9:      -7   9  2 -6  0  3  0 -2  0  1
10:    -11   5 11 -1 -6  0  3  0 -2  0  1
11:     11 -15 -3 11  0 -6  0  3  0 -2  0  1
...
n = 4: c(0, x)*a(4) + c(2, x)*a(2) + c(4, x)*a(0) = (+x^1)*3 +  (+x^1 + x^5)*(-2) + (+x^1 - x^3 + x^9)*1 = +2*x^1 - x^3 - 2*x^5 + 0*x^7 + x^9. Hence row n=4 is 2, -1, -2, 0, 1.
From A274660, row n = 4: c(4, x) = +x^1 - x^3 +x^9.
n = 4: P(4, x) = 2 - 1*x^1 - 2*x^2 + 1*x^4, that is the contribution of order q^4 to cn in the new variables is (2*cos(v)  - 1*cos(3*v) - 2*cos(5*v) + 1*cos(9*v))*q^4.

J. V. Armitage and W. F. Eberlein, Elliptic Functions, London Mathematical Society, Student Texts 67, Cambridge University Press, 2006.
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 15, Theorem 3.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, fourth edition, reprinted, 1958, Cambridge at the University Press.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 375, 16.23.2.
Wolfdieter Lang, Expansions for phase space coordinates for the plane pendulum

Cf. A000007, A099774, A274621, A274659, A274660, A275791.

T(n, m) = [x^(2*m+1)]Sum_{j=0..n} c(j, x)*a(n-j), with a(k) = A274621(k/2) if k is even and a(k) = 0 if k is odd, and c(j, x) = (-1)^j*Sum_{2*r+1 | 2*j+1} (-1)^r*x^(2*r+1) = Sum_{k=1..A099774(j+1)} sign(A274660(j, k))*x^(abs(A274660(j, k))), for j >= 0.

A144874 Coefficients of the series expansion of q^(-1/4) pi_q.

Original entry on oeis.org

1, 2, 0, 0, 1, -2, 1, 2, -3, 0, 2, 0, -1, 0, -1, 0, 4, -2, -2, 0, -1, 4, 1, -4, 0, 2, -2, 0, 2, 0, -1, 2, -1, -4, 2, 0, 2, 2, -2, 0, -2, -2, 3, 2, -3, 0, 4, -2, -2, 2, -2, 2, 0, -4, 0, 4, 3, -2, -1, -2, 0, 2, -2, -2, 2, 2, 2, 0, -4, 0, 2, -2, 1, 2, -3, -2, 4, 0, -2, 2, -2, 4, 0, -4, 2, -2, -2, 2, 2, -2, -1, 4, 1, -2, 2, -2, -4, 2, 0, 0, 2

Offset: 0

Eric W. Weisstein, Sep 23 2008

sign

From Peter Bala, Dec 12 2013: (Start)
The gamma function Gamma(x) has a q-extension or q-analog called the q-gamma function, denoted Gamma(q,x), defined by means of the product Gamma(q,x) := 1/(1-q)^(x-1)*( product{n >= 1} (1 - q^n)/(1 - q^(n+x-1)) ) when |q| < 1.
The gamma and q-gamma functions are related through the limiting process Gamma(x) = lim {q -> 1 from below} Gamma(q,x).
It is well known that the constant Pi = Gamma(1/2)^2. This suggests defining a function Pi(q), a q-analog of Pi, by putting Pi(q) = Gamma(q^2,1/2)^2 = (1 - q^2)*( product {n >= 1} (1 - q^(2*n))/(1 - q^(2*n-1)) )^2 = 1 + 2*q + q^4 - 2*q^5 + q^6 + .... This sequence gives the coefficients in the Maclaurin expansion of Pi(q).
Several classical formulas involving Pi have generalizations that involve the function Pi(q). See the Formula section below. (End)

			G.f. = 1 + 2*x + x^4 - 2*x^5 + x^6 + 2*x^7 - 3*x^8 + 2*x^10 + ...

R. Roy, Sources in the development of mathematics, Cambridge University Press 2011.
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.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
R. W. Gosper, Experiments and discoveries in q-trigonometry, Preprint.
R. W. Gosper, q-Trigonometry: Some Prefatory Afterthoughts
K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94
Eric Weisstein's World of Mathematics, q-Pi
Wikipedia, Leibniz formula for Pi
Wikipedia, q-analog
Wikipedia, q-gamma function
Wikipedia, Wallis product

Cf. A008441.

max = 100; pi[q_] := (1 - q^2)*q^(1/4)*Product[(1 - q^(2n))^2 / (1 - q^(2n - 1))^2, {n, 1, max}]; CoefficientList[ Series[ q^(-1/4)*pi[q], {q, 0, max}], q] (* Jean-François Alcover, Feb 07 2013 *)

From Peter Bala, Dec 12 2013: (Start)
Pi(q) = q^(1/4)*pi_q.
Pi(q) = (1 - q^2)*( Sum_{n >=0} q^(n*(n+1)/2) )^2.
Some q-analogs of classical formulas
= = = = = = = = = = = = = = = = = = =
Let [n] := 1 + q + q^2 + ... + q^(n-1) denote the q-analog of the natural number n.
(a) Wallis' formula Pi/2 = (2/1)*(2/3)*(4/3)*(4/5)*(6/5)*(6/7)* ....
q_analog: Pi(q)/[2] = ([2]/[1])*([2]/[3])*([4]/[3])*([4]/[5])*([6]/[5])*([6]/[7])* ....
(b) The Euler-Sylvester continued fraction Pi/2 = 1 + 1/(1 + 2/(1 + 6/(1 + 12/(1 + ...)))) (Roy 3.47 and 3.67).
q-analog: Pi(q)/[2] = 1 + q/(1 + q*[1]*[2]/(1 + q*[2]*[3]/(1 + q*[3]*[4]/(1 + ...)))).
(c) The Madhava-Leibniz series Pi/4 = 1 - 1/3 + 1/5 - 1/7 + ....
We have two q-analogs:
Pi(q^2)/[4] = 1/[1] - q/[3] + q^2/[5] - q^3/[7] + ...,
as well as
Pi(q)/[2] = sum {n in Z} (-1)^n*q^(n*(n+1))/[2*n+1].
(d) The result Pi^2/8 = sum {n >= 0} 1/(2*n+1)^2.
q-analog: Pi(q^2)^2/[2]^2 = (1 + q)/[1]^2 + q*(1 + q^3)/[3]^2 + q^2*(1 + q^5)/[5]^2 + ....
(e) The result Pi^4/96 = sum {n >= 0} 1/(2*n+1)^4.
q-analog: q*Pi(q^2)^4/[2]^4 = f(q)/[1]^4 + f(q^3)/[3]^4 + f(q^5)/[5]^4 + ..., where f(q) = q + 4*q^2 + q^3. (End)
a(n) = A008441(n) - A008441(n-2) for n > 1. - Seiichi Manyama, Jan 05 2022

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Julia

Maple

Mathematica

PARI

PARI

PARI

PARI

SageMath

Formula

Extensions

A008441 Number of ways of writing n as the sum of 2 triangular numbers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

PARI

PARI

PARI

Sage

Formula

Extensions

A366730 Expansion of g.f. A(x,y) satisfying 0 = Sum_{n=-oo..+oo} x^n * A(x,y)^n * (y - x^(n-1))^(n+1), as a triangle of coefficients T(n,k) of x^n*y^k in A(x,y), read by rows n >= 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A273225 Number of bipartitions of n wherein odd parts are distinct (and even parts are unrestricted).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

A295832 Expansion of Product_{k>=1} ((1 + x^(2*k-1))/(1 - x^(2*k)))^k.

Views

Author

Keywords

Crossrefs

A295832 Expansion of Product_{k>=1} ((1 + x^(2k-1))/(1 - x^(2k)))^k.