A001936 -id:A001936 - cp's OEIS frontend

A001935 Number of partitions with no even part repeated; partitions of n in which no parts are multiples of 4.

1, 1, 2, 3, 4, 6, 9, 12, 16, 22, 29, 38, 50, 64, 82, 105, 132, 166, 208, 258, 320, 395, 484, 592, 722, 876, 1060, 1280, 1539, 1846, 2210, 2636, 3138, 3728, 4416, 5222, 6163, 7256, 8528, 10006, 11716, 13696, 15986, 18624, 21666, 25169, 29190, 33808, 39104, 45164

Offset: 0

N. J. A. Sloane, Simon Plouffe, Robert G. Wilson v

Also number of partitions of n where no part appears more than three times.
a(n) satisfies Euler's pentagonal number (A001318) theorem, unless n is in A062717 (see Fink et al.).
Also number of partitions of n in which the least part and the differences between consecutive parts is at most 3. Example: a(5)=6 because we have [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1] and [1,1,1,1,1]. - Emeric Deutsch, Apr 19 2006
Equals A000009 convolved with its aerated variant, = polcoeff A000009 * A000041 * A010054 (with alternate signs). - Gary W. Adamson, Mar 16 2010
Equals left border of triangle A174715. - Gary W. Adamson, Mar 27 2010
The Cayley reference is actually to A083365. - Michael Somos, Feb 24 2011
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution of A000009 and A035457. - Vaclav Kotesovec, Aug 23 2015
Convolution inverse is A082303. - Michael Somos, Sep 30 2017
The g.f. in the form Sum_{n >= 0} x^(n*(n+1)/2) * Product_{k = 1..n} (1+x^k)/(1-x^k) = Sum_{n >= 0} x^(n*(n+1)/2) * Product_{k = 1..n} (1+x^k)/(1+x^k-2*x^k) == Sum_{n >= 0} x^(n*(n+1)/2) (mod 2). It follows that a(n) is odd iff n = k*(k + 1)/2 for some nonnegative integer k. Cf. A333374. - Peter Bala, Jan 08 2025

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 12*x^7 + 16*x^8 + 22*x^9 + ...
G.f. = q + q^9 + 2*q^17 + 3*q^25 + 4*q^33 + 6*q^41 + 9*q^49 + 12*q^57 + 16*q^65 + 22*q^73 + ...
a(5)=6 because we have [5], [4,1], [3,2], [3,1,1], [2,1,1,1] and [1,1,1,1,1].

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.5.2).
M. D. Hirschhorn, The Power of q, Springer, 2017. See ped page 303ff.
R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 241.
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).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
George E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573. (See Th. 9.)
George E. Andrews, Partition Identities for Two-Color Partitions, Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2021, Special Commemorative volume in honour of Srinivasa Ramanujan, 2021, 44, pp.74-80. hal-03498190. See Theorem 1.4 p. 75.
Riccardo Aragona, Roberto Civino, and Norberto Gavioli, A modular idealizer chain and unrefinability of partitions with repeated parts, arXiv:2301.06347 [math.RA], 2023.
Cristina Ballantine and Mircea Merca, Parity of sums of partition numbers and squares in arithmetic progressions, The Ramanujan Journal, 2016.
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129.]
S.-C. Chen, On the number of partitions with distinct even parts, Discrete Math., 311 (2011), 940-943.
A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008).
M. D. Hirschhorn and J. A. Sellers, A Congruence Modulo 3 for Partitions into Distinct Non-Multiples of Four, Article 14.9.6, Journal of Integer Sequences, Vol. 17 (2014).
Joro, Is "OEIS A001935 Number of partitions with no even part repeated" efficiently computable mod 4?
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. 15.
Mircea Merca, New relations for the number of partitions with distinct even parts, Journal of Number Theory 176 (July 2017), 1-12.
Alexander Patkowski, On some partitions where even parts do not repeat, Demonstratio Mathematica Volume 42, Issue 2 (Jun 2009), pp. 259-263.
Eric Weisstein's World of Mathematics, Partition Function b_k and Partition Function P.
Wikipedia, Glaisher's Theorem.

Cf. A000009, A000726, A001936, A035959, A035985, A042968, A061198, A061199, A070048, A081055, A081056, A083365, A098491, A098492, A219601.
Cf. A000041, A010054. - Gary W. Adamson, Mar 16 2010
Cf. A174715. - Gary W. Adamson, Mar 27 2010
Cf. A082303.
Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

a001935 = p a042968_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Sep 02 2012

g:=product((1+x^j)*(1+x^(2*j)),j=1..50): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=0..48); # Emeric Deutsch, Apr 19 2006
# second Maple program:
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
     `if`(irem(d, 4)=0, 0, d), d=divisors(j)), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Nov 24 2015

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, Pi/4, q^(1/2)] / (16 q)^(1/8), {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 4, n, 4}] / Product[ 1 - x^k, {k, n}], {x, 0, n}]; (* Michael Somos, Jul 08 2011 *)
CoefficientList[Series[Product[1+x^j+x^(2j)+x^(3j), {j,1,48}], {x,0,48}],x] (* Jean-François Alcover, May 26 2011, after Jon Perry *)
QP = QPochhammer; CoefficientList[QP[q^4]/QP[q] + O[q]^50, q] (* Jean-François Alcover, Nov 24 2015 *)
a[0] = 1; a[n_] := a[n] = Sum[a[n-j] DivisorSum[j, If[Divisible[#, 4], 0, #]&], {j, 1, n}]/n; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 19 2016, after Alois P. Heinz *)
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 4], 0, 2] ], {n, 0, 49}] (* Robert Price, Jul 28 2020 *)

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

{a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint( 8*n + 1) - 1)\2, prod(i=1, k, (1 + x^i) / (x^-i - 1), 1 + x * O(x^n))), n))}; /* Michael Somos, Jun 01 2004 */

{a(n)=polcoeff(exp(sum(m=1, n+1, x^m/(1+(-x)^m+x*O(x^n))/m)),n)} \\ Paul D. Hanna, Jul 24 2013

Euler transform of period 4 sequence [ 1, 1, 1, 0, ...].
Expansion of q^(-1/8) * eta(q^4) / eta(q) in powers of q. - Michael Somos, Mar 19 2004
Expansion of psi(-x) / phi(-x) = psi(x) / phi(-x^2) = psi(x^2) / psi(-x) = chi(x) / chi(-x^2)^2 = 1 / (chi(x) * chi(-x)^2) = 1 / (chi(-x) * chi(-x^2)) = f(-x^4) / f(-x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Jul 08 2011
G.f.: Product(j>=1, 1 + x^j + x^(2*j) + x^(3*j)). - Jon Perry, Mar 30 2004
G.f.: Product_{k>=1} (1+x^k)^(2-k%2). - Jon Perry, May 05 2005
G.f.: Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k-1)) = 1 + Sum_{k>0}(Product_{i=1..k} (x^i + 1) / (x^-i - 1)).
G.f.: Sum_{n>=0} ( x^(n*(n+1)/2) * Product_{k=1..n} (1+x^k)/(1-x^k) ). - Joerg Arndt, Apr 07 2011
G.f.: P(x^4)/P(x) where P(x) = Product_{k>=1} 1-x^k. - Joerg Arndt, Jun 21 2011
A083365(n) = (-1)^n a(n). Convolution square is A001936. a(n) = A098491(n) + A098492(n). a(2*n) = A081055(n). a(2*n + 1) = A081056(n).
G.f.: (1+ 1/G(0))/2, where G(k) = 1 - x^(2*k+1) - x^(2*k+1)/(1 + x^(2*k+2) + x^(2*k+2)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jul 03 2013
G.f.: exp( Sum_{n>=1} (x^n/n) / (1 + (-x)^n) ). - Paul D. Hanna, Jul 24 2013
a(n) ~ Pi * BesselI(1, sqrt(8*n + 1)*Pi/4) / (2*sqrt(8*n + 1)) ~ exp(Pi*sqrt(n/2)) / (4 * (2*n)^(3/4)) * (1 + (Pi/(16*sqrt(2)) - 3/(4*Pi*sqrt(2))) / sqrt(n) + (Pi^2/1024 - 15/(64*Pi^2) - 15/128) / n). - Vaclav Kotesovec, Aug 23 2015, extended Jan 14 2017
a(n) = (1/n)*Sum_{k=1..n} A046897(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
G.f. is a period 1 Fourier series which satisfies f(-1 / (256 t)) = 1/2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A082303. - Michael Somos, Sep 30 2017

More terms from James Sellers

A022567 Expansion of Product_{m>=1} (1+x^m)^2.

Original entry on oeis.org

1, 2, 3, 6, 9, 14, 22, 32, 46, 66, 93, 128, 176, 238, 319, 426, 562, 736, 960, 1242, 1598, 2048, 2608, 3306, 4175, 5248, 6570, 8198, 10190, 12622, 15589, 19190, 23552, 28830, 35190, 42842, 52034, 63040, 76198, 91904, 110604, 132832, 159216, 190464, 227417

Offset: 0

N. J. A. Sloane, Jun 14 1998

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number of partitions of n into distinct parts, with 2 types of each part. E.g., for n=4, we consider k and k* to be different versions of k and so we have 4, 4*, 31, 31*, 3*1, 3*1*, 22*, 211*, 2*11*, thus a(4)=9. - Jon Perry, Apr 04 2004
Number of partitions of n into odd parts, each part being of two kinds. E.g., a(3)=6 because we have 3, 3', 1+1+1, 1+1+1', 1+1'+1', 1'+1'+1'. - Emeric Deutsch, Mar 22 2005
Euler transform of period 2 sequence [2,0,2,0,...]. - Emeric Deutsch, Mar 22 2005
Equals A000041 convolved with A010054. - Gary W. Adamson, Jun 11 2009
The sum of the least gaps in all partitions of n. The "least gap" of a partition is the least positive integer that is not a part of the partition. Example: a(4) = 9 because the least gaps in [4], [3,1], [2,2], [2,1,1], and [1,1,1,1] are 1, 2, 1, 3, and 2, respectively. - Emeric Deutsch, May 18 2015
Number of 2-regular bipartitions of n. - N. J. A. Sloane, Oct 20 2019
The least gap is also known as the minimal excludant or mex; see Andrews and Newman. - George Beck, Dec 10 2020

			G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 9*x^4 + 14*x^5 + 22*x^6 + 32*x^7 + 46*x^8 + ...
G.f. = q + 2*q^13 + 3*q^25 + 6*q^37 + 9*q^49 + 14*q^61 + 22*q^73 + 32*q^85 + ...

P. J. Grabner, A. Knopfmacher, Analysis of some new partition statistics, Ramanujan J., 12, 2006, 439-454.
Kathiravan, T., and S. N. Fathima. "On L-regular bipartitions modulo L." The Ramanujan Journal 44.3 (2017): 549-558.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
George E. Andrews, David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
Cristina Ballantine, Mircea Merca, Bisected theta series, least r-gaps in partitions, and polygonal numbers, arXiv:1710.05960 [math.CO], 2017.
Helena Bergold, Lukas Egeling, and Hung. P. Hoang, Signotopes with few plus signs, arXiv:2411.19208 [math.CO], 2024. See p. 14.
J. Currie, N. Rampersad, Binary words avoiding xx^Rx and strongly unimodal sequences, JIS 18 (2015) #15.10.3.
Alejandro Erickson, Frank Ruskey, Enumerating maximal tatami mat coverings of square grids with v vertical dominoes, arXiv:1304.0070 [math.CO], 2013.
Alejandro Erickson and Mark Schurch, Monomer-dimer tatami tilings of square regions, arXiv preprint arXiv:1110.5103 [math.CO], 2011.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 852
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. 8.
Mircea Merca, A new look on the generating function for the number of divisors, Journal of Number Theory, Volume 149, April 2015, Pages 57-69.
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. See q(n)'.
Mbavhalelo Mulokwe and Konstantinos Zoubos, Free fermions, neutrality and modular transformations, arXiv:2403.08531 [hep-th], 2024.
Michael Somos, Introduction to Ramanujan theta functions
Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000009, A022597, A285221, A304048.
Cf. A010054. - Gary W. Adamson, Jun 11 2009
Column k=2 of A286335.
Number of r-regular bipartitions of n for r = 2,3,4,5,6: A022567, A328547, A001936, A263002, A328548.

Coefficients(&*[(1+x^m)^2:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 26 2018

A022567 := proc(n)
    local x,m;
    product((1+x^m)^2,m=1..n) ;
    expand(%) ;
    coeff(%,x,n) ;
end proc: # R. J. Mathar, Jun 18 2016

a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2]^-2, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 + q^k, {k, n}]^2, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
(QPochhammer[-1, x]^2/4 + O[x]^30)[[3]] (* Vladimir Reshetnikov, Sep 22 2016 *)
nmax = 50; poly = ConstantArray[0, nmax+1]; poly[[1]] = 1; poly[[2]] = 2; poly[[3]] = 1; Do[Do[Do[poly[[j+1]] += poly[[j-k+1]], {j, nmax, k, -1}]; , {p, 1, 2}], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 14 2017 *)

{a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n))^2, n))}; /* Michael Somos, Mar 21 2004 */

{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, Jun 03 2005 */

# uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 1, 0, 2)
a = EulerTransform(b)
print([a(n) for n in range(45)]) # Peter Luschny, Nov 11 2020

a(n) = p(n)+p(n-1)+p(n-3)+p(n-6)+...+p(n-k*(k+1)/2)+..., where p() is A000041(). E.g. a(8) = p(8)+p(7)+p(5)+p(2) = 22+15+7+2 = 46. - Vladeta Jovovic, Aug 09 2004
Expansion of q^(-1/12) * (eta(q^2) / eta(q))^2 in powers of q. - Michael Somos, Apr 27 2008
Expansion of chi(-q)^(-2) in powers of q where chi() is a Ramanujan theta function. - Michael Somos, Apr 27 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A022597. - Michael Somos, Apr 27 2008
G.f.: Product_{k>0} (1 + x^k)^2.
Convolution square of A000009. Convolution inverse of A022597. - Michael Somos, Apr 27 2008
Parity result: a(n) is even except when n is twice a generalized pentagonal number (i.e., of the form 2*A001318(m) for some m). - Peter Bala, Mar 19 2009
a(n) ~ exp(Pi * sqrt(2*n/3)) / (4 * 6^(1/4) * n^(3/4)) * (1 + (Pi/(12*sqrt(6)) - 3*sqrt(3/2)/(8*Pi)) / sqrt(n) + (Pi^2/1728 - 45/(256*Pi^2) - 5/64)/n). - Vaclav Kotesovec, Mar 05 2015, extended Jan 22 2017
a(0) = 1, a(n) = (2/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017
G.f.: exp(2*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

A079006 Expansion of q^(-1/4) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^2 in powers of q.

Original entry on oeis.org

1, -2, 5, -10, 18, -32, 55, -90, 144, -226, 346, -522, 777, -1138, 1648, -2362, 3348, -4704, 6554, -9056, 12425, -16932, 22922, -30848, 41282, -54946, 72768, -95914, 125842, -164402, 213901, -277204, 357904, -460448, 590330, -754368, 960948, -1220370

Offset: 0

Michael Somos, Dec 22 2002

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

The Lagrange series reversion of Sum_{n >= 1} a(n-1)*x^n is Sum_{n >= 1} A002103(n-1)*x^n. See the example in A002103. - Wolfdieter Lang, Jul 09 2016

			G.f. A(x) = 1 - 2*x + 5*x^2 - 10*x^3 + 18*x^4 - 32*x^5 + 55*x^6 - 90*x^7 + 144*x^8 + ...
G.f. B(q) = q * A(q^4) = q - 2*q^5 + 5*q^9 - 10*q^13 + 18*q^17 - 32*q^21 + 55*q^25 - 90*q^29 + ...

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
H. R. P. Ferguson, D. E. Nielsen and G. Cook, A partition formula for the integer coefficients of the theta function nome, Math. Comp., 29 (1975), 851-855.
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Dritter Teil, Springer-Verlag, 2012, eq. (8), p. 11 with eq. (2), p. 10.
Nicco, Conjectured identity of the product of two theta functions, Math Stackexchange, Sep 09 2015.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000700, A001936, A002103, A008441, A029839, A046897, A083365, A127391, A127392, A189925.

a[ n_] := SeriesCoefficient[ Product[(1 + x^(k + 1)) / (1 + x^k), {k, 1, n, 2}]^2, {x, 0, n}]; (* Michael Somos, Jul 08 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ[ q]}, SeriesCoefficient[ (m / 16 / q)^(1/4), {q, 0, n}]]; (* Michael Somos, Jul 08 2011 *)
QP = QPochhammer; s = (QP[q]*(QP[q^4]^2/QP[q^2]^3))^2 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 23 2015 *)
nmax = 50; CoefficientList[Series[Product[(1+x^(2*k))^4 / (1+x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 04 2016 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^4]^2 / QPochhammer[ -x]^2, {x, 0, n}]; (* Michael Somos, Apr 19 2017 *)

{a(n) = my(N, A); if( n<0, 0, N = (sqrtint(16*n + 1) + 1)\2; A = contfracpnqn( matrix(2, N, i, j, if( i==1, if( j<2, 1 + O(x^(N^2 + N)), (x^(j-1) + x^(3*j - 3))^2), 1 - x^(4*j - 2)))); polcoeff( A[2,1] / A[1,1], 4*n))}; /* Michael Somos, Sep 01 2005 */

{a(n) = my(A, m); if( n<0, 0, A = 1 + O(x); m = 1; while( m<=n, m*=2; A = subst(A, x, x^2); A = sqrt(A / (1 + 4 * x*A^2))); polcoeff(A, n))};

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

a(n) = (2/n)*Sum_{k=1..n} (-1)^k*A046897(k)*a(n-k). - Vladeta Jovovic, Dec 24 2002
Expansion of q^(-1/4) * (1/2) * k^(1/2) in powers of q, where k^2 is the parameter and q the Jacobi nome of elliptic functions.
Expansion of (1/(2*q)) * (1 - sqrt(k')) / (1 + sqrt(k')) in powers of q^4, where k'^2 is the complementary parameter and q the Jacobi nome of elliptic functions. See the Fricke reference.
Expansion of psi(x^2) / phi(x) = psi(x)^2 / phi(x)^2 = psi(x^2)^2 / psi(x)^2 = psi(-x)^2 / phi(-x^2)^2 = chi(-x)^2 / chi(-x^2)^4 = 1 / (chi(x)^2 * chi(-x^2)^2) = 1 / (chi(x)^4 * chi(-x)^2) = f(-x^4)^2 / f(x)^2 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 4 sequence [-2, 4, -2, 0, ...].
G.f. A(x) satisfies A(x)^2 = A(x^2) / (1 + 4 * x * A(x^2)^2). - Michael Somos, Mar 19 2004
Given g.f. A(x), then B(q) = q * A(q^4) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2 * (1 + 4 * v^2) - v. - Michael Somos, Jul 09 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 * (u6 + u2)^2 - u2*u6. - Michael Somos, Jul 09 2005
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 + x^(2*k-1)))^2 = (Product_{k>0} (1 - x^(4*k)) / (1 - (-x)^k))^2.
Expansion of continued fraction 1 / (1 - x^2 + (x^1 + x^3)^2 / (1 - x^6 + (x^2 + x^6)^2 / (1 - x^10 + (x^3 + x^9)^2 / ...))) in powers of x^4. - Michael Somos, Sep 01 2005
Given g.f. A(x), then B(q) = 2 * q * A(q^4) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (1 - u^4) * (1 - v^4) - (1 - u*v)^4 . - Michael Somos, Jan 01 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A189925.
Convolution inverse of A029839. Convolution square of A083365. a(n) = (-1)^n * A001936(n).
G.f.: 1/Q(0), where Q(k)= 1 - x^(k+1/2) + (x^((k+1)/4) + x^((3*k+3)/4))^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 02 2013
a(n) ~ (-1)^n * exp(Pi*sqrt(n)) / (2^(7/2)*n^(3/4)). - Vaclav Kotesovec, Jul 04 2016
Given g.f. A(x), and B(x) is the g.f. for A008441, then A(x) = B(x^2) / B(x) and A(x) * A(x^2) * A(x^4) * ... = 1 / B(x). - Michael Somos, Apr 20 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
a(n) = A208933(4*n+1) - A215348(4*n+1) (conjectured). - Thomas Baruchel, May 14 2018
A(x^4) = (1/(m*x)) * ( chi(x)^m - chi(-x)^m ) / ( chi(x)^m + chi(-x)^m ) at m = 2, where chi(x)  = Product_{i >= 0} (1 + x^(2*i+1)) is the g.f. of A000700. The formula gives generating functions related to A092869 when m = 1 and A001938 (also A093160) when m = 4. - Peter Bala, Sep 23 2023

A001938 Expansion of k/(4*q^(1/2)) in powers of q, where k defined by sqrt(k) = theta_2(0, q)/theta_3(0, q).

Original entry on oeis.org

1, -4, 14, -40, 101, -236, 518, -1080, 2162, -4180, 7840, -14328, 25591, -44776, 76918, -129952, 216240, -354864, 574958, -920600, 1457946, -2285452, 3548550, -5460592, 8332425, -12614088, 18953310, -28276968, 41904208, -61702876, 90304598, -131399624

Offset: 0

N. J. A. Sloane

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

k^2 is the parameter and q the Jacobi nome of elliptic functions. See, e.g., Fricke, p. 11, eq. (8) with p. 10. eq. (1). - Wolfdieter Lang, Jul 04 2016

			G.f. = 1 - 4*x + 14*x^2 - 40*x^3 + 101*x^4 - 236*x^5 + 518*x^6 - 1080*x^7 + ...
G.f. of B(q) = q * A(q^2): q - 4*q^3 + 14*q^5 - 40*q^7 + 101*q^9 - 236*q^11 + 518*q^13 - 1080*q^15 + ...

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, 1935, Oxford University Press, p. 385.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
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).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Dritter Teil, Springer-Verlag, 2012.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000122, A000700, A001936, A010054, A079006, A093160, A121373, A127393, A127931, A127932, A139820.

a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ -x, x^2] QPochhammer[ x^2, x^4])^4, {x, 0, n}]; (* Michael Somos, Sep 24 2011 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^4] / QPochhammer[ -x])^4, {x, 0, n}]; (* Michael Somos, Sep 24 2011 *)
a[ n_] := SeriesCoefficient[ (Product[ 1 - x^k, {k, 4, n, 4}] / Product[ 1 - (-x)^k, {k, n}])^4, {x, 0, n}]; (* Michael Somos, Sep 24 2011 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^(1/2)] / (2 EllipticTheta[ 3, 0, q]))^4, {q, 0, n + 1/2}]; (* Michael Somos, Sep 24 2011 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, 0, q^(1/2)])^4, {q, 0, n + 1/2}]; (* Michael Somos, Sep 24 2011 *)

{a(n) = my(A, A2, m); if( n<0, 0, n = 2*n + 1; A = x + O(x^3); m=2; while( mMichael Somos, Mar 26 2004 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^4, n))}; /* Michael Somos, Mar 26 2004 */

Expansion of (psi(x^2) / phi(x))^2 = (psi(x) / phi(x))^4 = (psi(x^2) / psi(x))^4 = (psi(-x) / psi(-x^2))^4 = (chi(-x) / chi(-x^2)^2)^4 = (chi(x)^2 * chi(-x))^-4 = (chi(x) * chi(-x^2))^-4 = (f(-x^4) / f(x))^4 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Feb 26 2012
G.f. A(x) satisfies 1 = (1 - 16 * x * A(x)^2) * (1 + 16 * x * A(-x)^2). - Michael Somos, Mar 26 2004
Expansion of q^(-1/2) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^4 in powers of q.
Euler transform of period 4 sequence [ -4, 8, -4, 0, ...].
Given g.f. A(x), then B(q) = q * A(q^2) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v - (u * (1 + 4*v))^2. - Michael Somos, Mar 26 2004
G.f. A(q) satisfies A(q) = sqrt(A(q^2)) / (1 + 4*q*A(q^2)); together with limit_{n->infinity} A(x^n) = 1 this gives a fast algorithm to compute the series. - Joerg Arndt, Aug 06 2011
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 + x^(2*k - 1)))^4.
a(n) = (-1)^n * A093160(n). Convolution square of A079006.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 1/4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A139820. - Michael Somos, Jun 04 2015
G.f.: ((Sum_{n >= 0} x^(n*(n+1))) / (1 + Sum_{n >= 1} x^(n^2)))^4 (from the sum representation of the Jacobi theta functions evaluated at vanishing argument). - Wolfdieter Lang, Jul 04 2016
a(n) ~ (-1)^n * exp(sqrt(2*n)*Pi) / (32 * 2^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

Edited by N. J. A. Sloane, Mar 31 2007

A007096 Expansion of theta_3 / theta_4.

Original entry on oeis.org

1, 4, 8, 16, 32, 56, 96, 160, 256, 404, 624, 944, 1408, 2072, 3008, 4320, 6144, 8648, 12072, 16720, 22976, 31360, 42528, 57312, 76800, 102364, 135728, 179104, 235264, 307672, 400704, 519808, 671744, 864960, 1109904, 1419456, 1809568, 2299832

Offset: 0

N. J. A. Sloane, Simon Plouffe

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number of partitions of 2n into parts with 2 types c, c* of each part. The even parts appears with multiplicity 1 for each type. The odd parts appears with multiplicity 2 (cc or c*c* but not cc*, that is, no mixing is allowed). E.g., a(4)=8 because of 44*, 22*, 211, 21*1*, 2*1*1*, 2*11, 111*1*. - Noureddine Chair, Jan 27 2005
a(n) is the number of pairs of overpartitions into odd parts where the sum of all parts is equal to n. - Jeremy Lovejoy, Aug 29 2020

			G.f. = 1 + 4*q + 8*q^2 + 16*q^3 + 32*q^4 + 56*q^5 + 96*q^6 + 160*q^7 + 256*q^8 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Shane Chern, Dennis Eichhorn, Shishuo Fu, and James A. Sellers, Convolutive sequences, I: Through the lens of integer partition functions, arXiv:2507.10965 [math.CO], 2025. See pp. 4, 8, 13.
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. 11.
Bernard L.S. Lin, Arithmetic properties of overpartition pairs into odd parts, Electronic J. Combin. 19, 2012, Paper 17.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Self-convolution of A080054. - Vladeta Jovovic, Mar 22 2005
Cf. A014969, A001936, A001938, A079006, A127391, A127392.
Cf. A022577, A080054, A097243, A123655.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] / EllipticTheta[ 4, 0, q], {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m)^(-1/4), {q, 0, n}]]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[( QPochhammer[ -q, q^2] / QPochhammer[ q, q^2])^2, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ (Product[ 1 - (-q)^k, {k, n}] / Product[ 1 - q^k, {k, n}])^2, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
nmax=60; CoefficientList[Series[Product[((1+x^(2*k+1))/(1-x^(2*k+1)))^2, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 28 2015 *)

{a(n) = my(A, B); if( n<0, 0, A = 1 + 4*x; for( k=2, n, B = A + x^2 * O(x^k); A += Pol(2 * subst(B, x, x^2)^2 - B - 1/B) / x / 8); polcoeff(A, n))}; /* Michael Somos, Jul 07 2005*/

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)))^2, n))}; /* Michael Somos, Jan 01 2006 */

Euler transform of period 4 sequence [4, -2, 4, 0, ...]. - Vladeta Jovovic, Mar 22 2005
Expansion of eta(q^2)^6 /(eta(q)^4 * eta(q^4)^2) in powers of q.
Expansion of phi(q) / phi(-q) = chi(q)^2 / chi(-q)^2 = psi(q)^2 / psi(-q)^2 = phi(-q^2)^2 / phi(-q)^2 = phi(q)^2 / phi(-q^2)^2 = chi(-q^2)^2 / chi(-q)^4 = chi(q)^4 / chi(-q^2)^2 = f(q)^2 / f(-q)^2 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (1 - u^4) * (1 - v^4) - (1 - u*v)^4. - Michael Somos, Jan 01 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A028939.
Expansion of Jacobian elliptic function 1 / sqrt(k') in powers of q. - see Fine.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 1 + u^2 - 2*u*v^2. - Michael Somos, Jul 07 2005
Unique solution to f(x^2)^2 = (f(x) + 1 / f(x)) / 2 and f(0)=1, f'(0) nonzero.
G.f.: theta_3 / theta_4 = (Sum_{k} x^k^2) / (Sum_{k} (-x)^k^2) = (Product_{k>0} (1 - x^(4*k - 2)) / ((1 - x^(4*k - 1)) * (1 - x^(4*k - 3)))^2)^2.
A097243(n) = a(4*n). 8*A022577(n) = a(4*n + 2). a(n) = 4*A123655(n) if n>0. Convolution square of A080054.
Empirical: sum(exp(-Pi)^(n-1)*a(n),n=1..infinity) = 2^(1/4). - Simon Plouffe, Feb 20 2011
Empirical : sum(exp(-Pi*sqrt(2))^(n-1)*(-1)^(n+1)*a(n),n=1..infinity) = (-2+2*2^(1/2))^(1/4). - Simon Plouffe, Feb 20 2011
Empirical : sum(exp(-2*Pi)^(n-1)*a(n),n=1..infinity) = 1/2*(8+6*2^(1/2))^(1/4). - Simon Plouffe, Feb 20 2011
a(n) ~ exp(Pi*sqrt(n)) / (4*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Aug 28 2015
G.f.: exp(4*Sum_{k>=1} sigma(2*k - 1)*x^(2*k-1)/(2*k - 1)). - Ilya Gutkovskiy, Apr 19 2019

A002103 Coefficients of expansion of Jacobi nome q in certain powers of (1/2)*(1 - sqrt(k')) / (1 + sqrt(k')).

Original entry on oeis.org

1, 2, 15, 150, 1707, 20910, 268616, 3567400, 48555069, 673458874, 9481557398, 135119529972, 1944997539623, 28235172753886, 412850231439153, 6074299605748746, 89857589279037102, 1335623521633805028

Offset: 0

N. J. A. Sloane

The Fricke reference has equation q^(1/4) = (sqrt(k) / 2) + 2(sqrt(k) / 2)^5 + 15(sqrt(k) / 2)^9 + 150(sqrt(k) / 2)^13 + 1707(sqrt(k) / 2)^17 + ... - Michael Somos, Jul 13 2013
a(n)^(1/n) tends to 16. - Vaclav Kotesovec, Jul 02 2016
a(n-1) appears in the expansion of the Jacobi nome q as q = x*Sum_{n >= 1} a(n-1)*x^(4*n) with x = (1/2)*(1 - sqrt(k')) / (1 + sqrt(k')), with the complementary modulus k' of elliptic functions. See, e.g., the Fricke, Kneser and Tricomi references, and the g.f. with example below. - Wolfdieter Lang, Jul 09 2016
The King-Canfield (1992) reference shows how this sequence is used in real life - it is one of the ingredients in solving the general quintic equation using elliptic functions. - N. J. A. Sloane, Dec 24 2019

			G.f. = 1 + 2*x + 15*x^2 + 150*x^3 + 1707*x^4 + 20910*x^5 + 268616*x^6 + 3567400*x^7 + ...
Jacobi nome q = x + 2x^5 + 15x^9 + 150x^13 + ... where x = q - 2q^5 + 5q^9 - 10q^13 + ... coefficients from A079006.
The series reversion of q = x + 2*x^5 + 15*x^9 + 150*x^13 + 1707*x^17 + ... equals (x + x^9 + x^25 + x^49 + ...)/(1 + 2*x^4 + 2*x^16 + 2*x^36 + 2*x^64 + ...).

King, R. B., and E. R. Canfield. "Icosahedral symmetry and the quintic equation." Computers & Mathematics with Applications 24.3 (1992): 13-28. See Eq. (4.28).
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).
F. Tricomi, Elliptische Funktionen (German translation by M. Krafft of: Funzioni ellittiche), Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1948, p. 176, eq. (3.88).
Z. X. Wang and D. R. Guo, Special Functions, World Scientific Publishing, 1989, page 512.

Vaclav Kotesovec, Table of n, a(n) for n = 0..800 (terms 0..200 from Vincenzo Librandi)
J. N. Bramhall, An iterative method for inversion of power series, Comm. ACM 4 1961 317-318.
H. R. P. Ferguson, D. E. Nielsen and G. Cook, A partition formula for the integer coefficients of the theta function nome, Math. Comp., 29 (1975), 851-855.
H. E. Fettis, Note on the computation of Jacobi's Nome and its inverse, Computing, 4 (1969), 202-206.
A. Fletcher, Guide to tables of elliptic functions, Math. Tables Other Aids Computation, 3 (1948), 229-281, Section III, p. 234. MR0030295 (10,741b)
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Dritter Teil, Springer-Verlag, 2012, p. 3, eq. (6)
C. Hermite, Annotated scan of a page from the Oeuvres, together with a page from Math. Tables Aids Comp., Vol. 3, 1948 that refers to it.
A. Kneser, Neue Untersuchung einer Reihe aus der Theorie der elliptischen Funktionen, J. reine u. angew. Math. 157, 1927, 209 - 218, p.218.
A. N. Lowan, G. Blanch and W. Horenstein, On the inversion of the q-series associated with Jacobian elliptic functions, Bull. Amer. Math. Soc., 48 (1942), 737-738.
H. P. Robinson, Letter to N. J. A. Sloane, Oct 07, 1976
R. E. Shafer, Review of Fettis (1969), Computing Reviews, July 1970, page 401 [Annotated scanned copy]

Cf. A001936, A002639, A079006.

max = 18; A079006[n_] := SeriesCoefficient[ Product[(1+x^(k+1)) / (1+x^k), {k, 1, n, 2}]^2, {x, 0, n}]; A079006[0] = 1; sq = Series[ Sum[ A079006[n]*q^(4n+1), {n, 0, max}], {q, 0, 4max}]; coes = CoefficientList[ InverseSeries[ sq, x], x]; a[n_] := coes[[4n + 2]]; Table[a[n], {n, 0, max-1}] (* Jean-François Alcover, Nov 08 2011, after Michael Somos *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ (EllipticNomeQ[ 16 x] / x)^(1/4), {x, 0, n}]]; (* Michael Somos, Jul 13 2013 *)
a[ n_] := With[{m = 4 n + 1}, If[ n < 0, 0, SeriesCoefficient[ InverseSeries[ Series[ q (QPochhammer[ q^16] / QPochhammer[-q^4])^2, {q, 0, m}], x], {x, 0, m}]]]; (* Michael Somos, Jul 13 2013 *)
a[ n_] := With[{m = 4 n + 1}, SeriesCoefficient[ InverseSeries[ Series[ 1/2 EllipticTheta[ 2, 0, x^4] / EllipticTheta[ 3, 0, x^4], {x, 0, m}]], {x, 0, m}]]; (* Michael Somos, Apr 14 2015 *)

{a(n) = my(A); if( n<0, 0, n = 4*n + 1; A = O(x^n); polcoeff( serreverse( x * (eta(x^4 + A) * eta(x^16 + A)^2 / eta(x^8 + A)^3)^2), n))};

{a(n)=local(A,N=sqrtint(n)+1); A=serreverse(sum(n=1,N,x^((2*n-1)^2))/(1+2*sum(n=1,N,x^(4*n^2)) +O(x^(4*n+4)))); polcoeff(A,4*n+1)} \\ Paul D. Hanna, Jan 07 2014

a(n) = Sum {1<=k<=n} (-1)^k Sum { (4n+k)! C_1^b_1 ... C_n^b_n / (4n+1)! b_1! ... b_n! }, where the inner sum is over all partitions k = b_1 + ... + b_n, n = Sum i*b_i, b_i >= 0 and C_0=1, C_1=-2, C_2=5, C_3=-10 ... is given by (-1)^n*A001936(n).

G.f.: Series_Reversion( (theta_3(x) - theta_3(-x)) / (4*theta_3(x^4)) ) = Sum_{n>=0} a(n)*x^(4*n+1), where theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2). - Paul D. Hanna, Jan 07 2014

A092877 Expansion of (eta(q^4) / eta(q))^8 in powers of q.

Original entry on oeis.org

1, 8, 44, 192, 718, 2400, 7352, 20992, 56549, 145008, 356388, 844032, 1934534, 4306368, 9337704, 19771392, 40965362, 83207976, 165944732, 325393024, 628092832, 1194744096, 2241688744, 4152367104, 7599231223, 13749863984

Offset: 1

Michael Somos, Mar 19 2004

nonn

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

			G.f. = q + 8*q^2 + 44*q^3 + 192*q^4 + 718*q^5 + 2400*q^6 + 7352*q^7 + 20992*q^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
Kevin Acres, David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A001935, A001936, A005798, A014103, A093160, A124972.

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ -InverseEllipticNomeQ[ -x] / 16, {x, 0, n}]]; (* Michael Somos, Jun 13 2011 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ With[ {lambda = ModularLambda[ Log[x] / ( Pi I)]}, lambda / (16 * (1 - lambda))], {x, 0, n}]]; (* Michael Somos, Jun 13 2011 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^4] / QPochhammer[ q])^8, {q, 0, n}]; (* Michael Somos, Aug 09 2015 *)
a[1] = 1; a[n_] := a[n] = (8/(n-1))*Sum[DivisorSum[k, Identity, Mod[#, 4] != 0&]*a[n-k], {k, 1, n-1}]; Array[a, 26] (* Jean-François Alcover, Mar 01 2018, after Seiichi Manyama *)
eta[q_]:= q^(1/6) QPochhammer[q]; a[n_]:=SeriesCoefficient[(eta[q^4] / eta[q])^8, {q, 0, n}]; Table[a[n], {n, 4, 35}] (* Vincenzo Librandi, Oct 18 2018 *)

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

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^4 + A) / eta(x + A))^8, n))};

Expansion of q * (psi(-q) / phi(-q))^8 = q * (psi(q^2) / psi(-q))^8 = q * (psi(q) / phi(-q^2))^8 = q * (psi(q^2) / phi(-q))^4 = q * (chi(q) / chi(-q^2)^2)^8 = q / (chi(-q) * chi(-q^2))^8 = q / (chi(q) * chi(-q)^2)^8 = q * (f(-q^4) / f(-q))^8 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Jun 13 2011
Euler transform of period 4 sequence [ 8, 8, 8, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - 16*u*v - 16*v^2 - 256*u*v^2.
G.f.: x * (Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k - 1)))^8.
G.f.: theta_2^4 / (16*theta_4^4) = lambda / (16 * (1 - lambda)).
G.f.: exp( Integral theta_3(x)^4/x dx ). - Paul D. Hanna, May 03 2010
a(n) = (-1)^n * A005798(n).
a(2*n) = 8 * A014103(n). - Michael Somos, Aug 09 2015
Convolution inverse of A124972, 8th power of A001935, 4th power of A001936, square of A093160. - Michael Somos, Aug 09 2015
a(n) ~ exp(2*Pi*sqrt(n))/(512*n^(3/4)). - Vaclav Kotesovec, Sep 07 2015
a(1) = 1, a(n) = (8/(n-1))*Sum_{k=1..n-1} A046897(k)*a(n-k) for n > 1. - Seiichi Manyama, Apr 01 2017

A296044 a(n) = [x^n] Product_{k>=1} ((1 + x^(2k))/(1 - x^(2k-1)))^n.

Original entry on oeis.org

1, 1, 5, 22, 101, 481, 2330, 11425, 56549, 281911, 1413465, 7120136, 36006362, 182681916, 929461993, 4740491107, 24229115109, 124069449335, 636376573943, 3268955179686, 16814509004601, 86593280920756, 446437797872016, 2303948443259841, 11900990745759578, 61526182236027756

Offset: 0

Ilya Gutkovskiy, Dec 03 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..500

Main diagonal of A296068.

Cf. A001935, A001936, A001937, A001939, A001940, A001941, A092877, A093160, A295831, A296043, A296045.

Table[SeriesCoefficient[Product[((1 + x^(2 k))/(1 - x^(2 k - 1)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[Product[((1 - x^(4 k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[(EllipticTheta[2, 0, x]/EllipticTheta[2, Pi/4, x^(1/2)]/(16 x)^(1/8))^n, {x, 0, n}], {n, 0, 25}]
(* Calculation of constant d: *) With[{k = 4}, 1/r /. FindRoot[{s == QPochhammer[(r*s)^k] / QPochhammer[r*s], k*(-(s*QPochhammer[r*s]*(Log[1 - (r*s)^k] + QPolyGamma[0, 1, (r*s)^k]) / Log[(r*s)^k]) + (r*s)^k * Derivative[0, 1][QPochhammer][(r*s)^k, (r*s)^k]) == s*QPochhammer[r*s] + s^2*(-(QPochhammer[r*s]*(Log[1 - r*s] + QPolyGamma[0, 1, r*s]) / (s*Log[r*s])) + r*Derivative[0, 1][QPochhammer][r*s, r*s])}, {r, 1/5}, {s, 1}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)

a(n) = [x^n] Product_{k>=1} ((1 - x^(4*k))/(1 - x^k))^n.

a(n) ~ c * d^n / sqrt(n), where d = 5.2749356339591798618290252741994029798069148326559... and c = 0.2726256757090475625917361066565981461455343437... - Vaclav Kotesovec, Dec 05 2017

A001937 Expansion of (psi(x^2) / psi(-x))^3 in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 3, 9, 22, 48, 99, 194, 363, 657, 1155, 1977, 3312, 5443, 8787, 13968, 21894, 33873, 51795, 78345, 117312, 174033, 255945, 373353, 540486, 776848, 1109040, 1573209, 2218198, 3109713, 4335840, 6014123, 8300811, 11402928, 15593702, 21232521, 28790667, 38884082

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

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

The Cayley reference is actually to A187053. - Michael Somos, Jul 26 2012

			1 + 3*x + 9*x^2 + 22*x^3 + 48*x^4 + 99*x^5 + 194*x^6 + 363*x^7 + 657*x^8 + ...
q^3 + 3*q^11 + 9*q^19 + 22*q^27 + 48*q^35 + 99*q^43 + 194*q^51 + 363*q^59 + ...

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
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).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A001936, A079006, A001935, A083365, A001938, A093160, A001939, A001940, A001941.

g100:= mul((1+x^(2*k))/(1-x^(2*k-1)),k=1..50)^3:
S:= series(g100,x,101):
seq(coeff(S,x,j),j=0..100); # Robert Israel, Nov 30 2015

CoefficientList[ Series[Product[(1 - x^k)^(-3*Boole[Mod[k, 4] != 0]), {k, 1, 101}], {x, 0, 100}], x] (* Olivier GERARD, May 06 2009 *)
QP = QPochhammer; s = (QP[q^4]/QP[q])^3 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A) / eta(x + A))^3, n))} /* Michael Somos, Mar 06 2011 */

Expansion of q^(-3/8) * (eta(q^4) / eta(q))^3 in powers of q. - Michael Somos, Jul 26 2012
Euler transform of period 4 sequence [ 3, 3, 3, 0, ...]. - Michael Somos, Mar 06 2011
Convolution cube of A001935. A187053(n) = (-1)^n * a(n). - Michael Somos, Mar 06 2011
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k-1)))^3.
a(n) ~ 3^(1/4) * exp(sqrt(3*n/2)*Pi) / (16*2^(3/4)*n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

Corrected and extended by Simon Plouffe

Checked and more terms from Olivier GERARD, May 06 2009

A098613 Expansion of psi(x^2) / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 3, 4, 7, 10, 17, 23, 35, 48, 69, 93, 131, 173, 236, 310, 413, 536, 704, 903, 1170, 1489, 1904, 2403, 3044, 3811, 4784, 5951, 7409, 9157, 11325, 13912, 17095, 20891, 25519, 31029, 37708, 45632, 55184, 66495, 80050, 96064, 115173, 137680, 164425, 195860

Offset: 0

Michael Somos, Sep 17 2004

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This sequence convolved with A000009 gives A001936. - Gary W. Adamson, Mar 24 2011
a(n) is the number of partitions of n in which each odd part can occur any number of times but each even part is of two kinds and each kind can occur at most once. - Michael Somos, Dec 01 2019

			G.f. = 1 + x + 3*x^2 + 4*x^3 + 7*x^4 + 10*x^5 + 17*x^6 + 23*x^7 + ...
G.f. = q^5 + q^29 + 3*q^53 + 4*q^77 + 7*q^101 + 10*q^125 + 17*q^149 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A029552, A083906, A143161.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] / (2 x^(1/4) QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, Oct 29 2013 *)
a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x, x^2] QPochhammer[ x^2, x^4]^2), {x, 0, n}]; (* Michael Somos, Oct 29 2013 *)
nmax = 40; CoefficientList[ Series[Product[(1 + x^k) * (1 + x^(2*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^2, x^2]^3, {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x]^2 QPochhammer[ -x, x]^3, {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ -x^2, x^2]^2, {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(4*n+1)-1)\2, x^(k^2+k)) / eta(x + x * O(x^n)), n))};

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

{a(n) = if( n<0,0, polcoeff( sum(k=0, 2*n+1, prod(i=1, k, (1 - x^(2*n+2 - i)) / (1 - x^i))) / 2, n^2))};

Expansion of chi(x) / chi(-x^2)^3 = 1 / (chi(-x)* chi(-x^2)^2) = 1 / (chi(x)^2 * chi(-x)^3) in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Sep 07 2015
Expansion of q^(-5/24) * eta(q^4)^2 / (eta(q) * eta(q^2)) in powers of q.
Euler transform of period 4 sequence [1, 2, 1, 0, ...].
G.f. A(x) is the limit of x^(n^2+n) * P_{2*n+1}(1/x)/2 where P_n(q) = Sum_{k=0..n} C(n, k; q) and C(n, k; q) is the q-binomial coefficients. See A083906 for P_n(q).
G.f.: (Sum_{k>0} x^(k^2-k)) / (Product_{k>0} (1 - x^k)).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(11/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 8^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A143161. - Michael Somos, Sep 07 2015
G.f.: Product_{k>=1} (1 + x^(2*k))^2 / (1 - x^(2*k-1)). - Michael Somos, Dec 01 2019

cp's OEIS Frontend

A001935 Number of partitions with no even part repeated; partitions of n in which no parts are multiples of 4.

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A022567 Expansion of Product_{m>=1} (1+x^m)^2.

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A079006 Expansion of q^(-1/4) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^2 in powers of q.

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A001938 Expansion of k/(4*q^(1/2)) in powers of q, where k defined by sqrt(k) = theta_2(0, q)/theta_3(0, q).

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A007096 Expansion of theta_3 / theta_4.

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A296044 a(n) = [x^n] Product_{k>=1} ((1 + x^(2k))/(1 - x^(2k-1)))^n.