A080054 -id:A080054 - cp's OEIS frontend

A169975 Expansion of Product_{i>=0} (1 + x^(4*i+1)).

1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 3, 2, 0, 1, 3, 3, 1, 1, 4, 4, 1, 1, 4, 5, 2, 1, 5, 7, 3, 1, 5, 8, 5, 2, 6, 10, 6, 2, 6, 12, 9, 3, 7, 14, 11, 4, 7, 16, 15, 6, 8, 19, 18, 8, 9, 21, 23, 11, 10, 24, 27, 14, 11, 27, 34, 19, 13, 30, 39, 24, 15, 33, 47, 31, 18, 37, 54, 38

Offset: 0

N. J. A. Sloane, Aug 29 2010

nonn

Number of partitions into distinct parts of the form 4*k+1.
In general, if a > 0, b > 0, GCD(a,b) = 1 and g.f. = Product_{k>=0} (1 + x^(a*k + b)), then a(n) ~ exp(Pi*sqrt(n/(3*a))) / (2^(1 + b/a) * (3*a)^(1/4) * n^(3/4)) [Meinardus, 1954]. - Vaclav Kotesovec, Aug 26 2015
Convolution of A147599 and A169975 is A000700. - Vaclav Kotesovec, Jan 18 2017

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Günter Meinardus, Über Partitionen mit Differenzenbedingungen, Mathematische Zeitschrift (1954/55), Volume: 61, page 289-302

Cf. A000700, A261612, A280454, A280456, A280457.
Cf. A015128, A080054, A261610, A261611.
Cf. A000041, A000009, A035382, A035451.
Cf. A170966-A170975.
Cf. A262928, A147599, A281243, A281244, A281245.

nmax = 200; CoefficientList[Series[Product[(1 + x^(4*k+1)), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 26 2015 *)
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 4] == 1, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 18 2017 *)

G.f.: Sum_{n>=0} (x^(2*n^2 - n) / Product_{k=1..n} (1 - x^(4*k))). - Joerg Arndt, Mar 10 2011
G.f.: G(0)/x where G(k) = 1 - 1/(1 - 1/(1 - 1/(1+(x)^(4*k+1))/G(k+1) )); (recursively defined continued fraction, see A006950). - Sergei N. Gladkovskii, Jan 28 2013
a(n) ~ exp(Pi*sqrt(n)/(2*sqrt(3))) / (2^(7/4) * 3^(1/4) * n^(3/4)) * (1 - (3*sqrt(3)/(4*Pi) + Pi/(192*sqrt(3))) / sqrt(n)). - Vaclav Kotesovec, Aug 26 2015, extended Jan 18 2017

A261612 Expansion of Product_{k>=0} (1 + x^(3*k+1)).

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 3, 3, 2, 4, 4, 2, 4, 5, 3, 5, 7, 4, 5, 8, 6, 7, 10, 7, 7, 12, 10, 9, 14, 12, 10, 16, 16, 13, 19, 19, 15, 22, 24, 19, 25, 28, 22, 29, 35, 28, 33, 40, 33, 38, 48, 41, 44, 55, 48, 51, 66, 59, 58, 74, 69

Offset: 0

Vaclav Kotesovec, Aug 26 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000009, A000700, A035382, A169975.

Cf. A015128, A080054, A261610, A261611, A262928.

nmax = 100; CoefficientList[Series[Product[(1 + x^(3*k+1)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 3] == 1, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}];], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 13 2017 *)

a(n) ~ exp(Pi*sqrt(n)/3) / (2^(4/3) * sqrt(3) * n^(3/4)) * (1 - (Pi/144 + 9/(8*Pi)) / sqrt(n)). - Vaclav Kotesovec, Aug 26 2015, extended Jan 16 2017

G.f.: Sum_{k>=0} x^(k*(3*k - 1)/2) / Product_{j=1..k} (1 - x^(3*j)). - Ilya Gutkovskiy, Nov 24 2020

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

A098151 Number of partitions of 2*n with no part divisible by 3 and all odd parts occurring with even multiplicities.

Original entry on oeis.org

1, 2, 4, 6, 10, 16, 24, 36, 52, 74, 104, 144, 198, 268, 360, 480, 634, 832, 1084, 1404, 1808, 2316, 2952, 3744, 4728, 5946, 7448, 9294, 11556, 14320, 17688, 21780, 26740, 32736, 39968, 48672, 59122, 71644, 86616, 104484, 125768, 151072, 181104, 216684

Offset: 0

Noureddine Chair, Aug 29 2004

There are no partitions of 2n+1 in which all odd parts occur with even multiplicity. - Michael Somos, Apr 15 2012
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(n) is also the number of Schur overpartitions of n, i.e., the number of overpartitions of n where adjacent parts differ by at least 3 if the smaller is overlined or divisible by 3 and adjacent parts differ by at least 6 if the smaller is overlined and divisible by 3. - Jeremy Lovejoy, Mar 23 2015
Let A(q) denote the g.f. of this sequence. Let m be a nonzero integer. The simple continued fraction expansions of the real numbers A(1/(2*m)) and A(1/(2*m+1)) may be predictable. For a given positive integer n, the sequence of the n-th partial denominators of the continued fractions are conjecturally polynomial or quasi-polynomial in m for m sufficiently large. An example is given below. Cf. A080054. - Peter Bala, Jun 09 2025

			a(4)=10 because 8 = 4+4 = 4+2+2=2+2+2+2 = 2+2+2+1+1 = 2+2+1+1+1+1 = 4+2+1+1 = 4+1+1+1+1 = 2+1+1+1+1+1+1 = 1+1+1+1+1+1+1+1.
G.f. = 1 + 2*q + 4*q^2 + 6*q^3 + 10*q^4 + 16*q^5 + 24*q^6 + 36*q^7 + 52*q^8 + ...
From _Peter Bala_, Jun 09 2025: (Start)
G.f.: A(q) = f(q, q^2) / f(-q, -q^2).
Simple continued fraction expansions of A(1/(2*m)):
m =  2  [1;  1   9  1    5    8    45   4  1  2  1  1  1  3  3   2  2 ...]
m =  3  [1;  2  13  1   14   12   133   8  1  1  7  2  1  2  2   1  1 ...]
m =  4  [1;  3  17  1   27   16   297  12  2  2  1  1  1  2  2   2  2 ...]
m =  5  [1;  4  21  1   44   20   561  16  2  1  7  3  3  2  2  25  8 ...]
m =  6  [1;  5  25  1   65   24   949  20  3  2  1  1  1  3  4   2  1 ...]
m =  7  [1;  6  29  1   90   28  1485  24  3  1  7  4  5  2  1   1  6 ...]
m =  8  [1;  7  33  1  119   32  2193  28  4  2  1  1  1  4  6   2  1 ...]
m =  9  [1;  8  37  1  152   36  3097  32  4  1  7  5  7  2  1   1  3 ...]
m = 10  [1;  9  41  1  189   40  4221  36  5  2  1  1  1  5  8   2  1 ...]
...
The sequence of the 4th partial denominators [5, 14, 27, 44, ...] appears to be given by the polynomial (2*m + 1)*(m - 1) for m >= 2.
The sequence of the 6th partial denominators [45, 133, 297, 561, ...] appears to be given by the polynomial (2*m + 1)*(2*m^2 + 1) for m >= 2. (End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
George E. Andrews, 4-Shadows in q-Series and the Kimberling Index, Preprint, May 15, 2016.
Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011, 2004.
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, 14-16.
Byungchan Kim and Eunmi Kim, Partitions weighted by the number of two types of parts, Bull. Korean Math. Soc. (2024) Vol. 61, No. 6, 1677-1684. See p. 1679.
Jeremy Lovejoy, A theorem on seven-colored overpartitions and its applications, Int. J. Number Theory. 1 (2005) 215-224
Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
Andrew Sills, Towards an Automation of the Circle Method, Gems in Experimental Mathematics in Contemporary Mathematics, 2010, formula S24.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A015128, A001318, A080054, A080995, A010815, A103260, A215594.

series(product((1+x^k+x^(2*k))/(1-x^k+x^(2*k)),k=1..150),x=0,100);
# alternative program using expansion of f(q, q^2) / f(-q, -q^2):
with(gfun): series( add(x^(n*(3*n-1)/2),n = -8..8)/add((-1)^n*x^(n*(3*n-1)/2), n = -8..8), x, 100): seriestolist(%); # Peter Bala, Feb 05 2021

a[ n_] := SeriesCoefficient[ QPochhammer[ q^2] QPochhammer[ q^3]^2 / (QPochhammer[ q]^2 QPochhammer[ q^6]), {q, 0, n}] (* Michael Somos, Oct 23 2013 *)
nmax = 50; CoefficientList[Series[Product[(1+x^(3*k-1)) * (1+x^(3*k-2)) / ((1-x^(3*k-1)) * (1-x^(3*k-2))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 / (eta(x + A)^2 * eta(x^6 + A)), n))} /* Michael Somos, Dec 04 2004 */

Expansion of phi(-q^3) / phi(-q) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Apr 15 2012
Expansion of f(q, q^2) / f(-q, -q^2) in powers of q where f(,) is the Ramanujan two-variable theta function. - Michael Somos, Apr 15 2012
Expansion of eta(q^2) * eta(q^3)^2 / (eta(q)^2 * eta(q^6)) in powers of q.
G.f. = (Sum_{n = -oo..oo} (-1)^n*q^(3*n^2)) / (Sum_{n = -oo..oo} (-1)^n*q^(n^2)). - N. J. A. Sloane, Aug 09 2016
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 + u^2) * (u^2 + v^4) - 4 * u^2*v^4. - Michael Somos, Apr 15 2012
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^3 - v + 3 * u*v^2 - 3 * u^2*v^3. - Michael Somos, Dec 04 2004
Euler transform of period 6 sequence [2, 1, 0, 1, 2, 0, ...]. - Vladeta Jovovic, Sep 24 2004
Taylor series of product_{k=1..inf}(1+x^k+x^(2*k))/(1-x^k+x^(2*k))= product_{k=1..inf}(1+x^k)(1-x^(3k))/((1-x^k)(1+x^(3k)))=Theta_4(0, x^3)/theta_4(0, x)
a(n) ~ Pi * BesselI(1, Pi*sqrt(2*n/3)) / (3*sqrt(2*n)) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/4) * 3^(3/4) * n^(3/4)) * (1 - 3*sqrt(3)/(8*Pi*sqrt(2*n)) - 45/(256*Pi^2*n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 09 2017
Convolution of A000726 and A003105. - R. J. Mathar, Nov 17 2017
From Peter Bala, Jun 09 2025: (Start)
G.f.: A(q) = Sum_{n = -oo..oo} q^(n*(3*n+1)/2) / Sum_{n = -oo..oo} (-1)^n * q^(n*(3*n+1)/2).
Recurrences:
a(n) - a(n-1) - a(n-2) + a(n-5) + a(n-7) - a(n-12) - a(n-15) + + - - ... = f(n), where [0, 1, 2, 5, 7, 12, 15, ...] is the sequence of generalized pentagonal numbers A001318, a(n) is set equal to 0 for negative n and f(n) = 1 if n is a generalized pentagonal number, otherwise f(n) = 0 (see A080995). Compare with the recurrence for the partition function p(n) = A000041(n).
a(n) - 2*a(n-1) + 2*a(n-4) - 2*a(n-9) + 2*a(n-16) - 2*a(n-25) + - ... = g(n), where g(n) = 2*(-1)^k if n is of the form 3*(k^2), otherwise g(n) = 0. (End)

A261610 Expansion of Product_{k>=0} (1 + x^(3k+1))/(1 - x^(3k+1)).

Original entry on oeis.org

1, 2, 2, 2, 4, 6, 6, 8, 12, 14, 16, 22, 28, 32, 40, 50, 58, 70, 86, 100, 118, 144, 168, 194, 232, 272, 312, 366, 428, 490, 568, 660, 754, 866, 1000, 1140, 1300, 1492, 1696, 1924, 2196, 2490, 2812, 3192, 3610, 4062, 4588, 5174, 5806, 6530, 7342, 8218, 9208

Offset: 0

Vaclav Kotesovec, Aug 26 2015

nonn

Cf. A015128, A080054, A261611.

nmax = 60; CoefficientList[Series[Product[(1 + x^(3*k+1))/(1 - x^(3*k+1)), {k, 0, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(n/3)) * Gamma(1/3) / (2^(5/3) * 3^(1/3) * Pi^(2/3) * n^(2/3)).

A320968 Expansion of (Product_{k>0} theta_3(q^k)/theta_4(q^k))^(1/2), where theta_3() and theta_4() are the Jacobi theta functions.

Original entry on oeis.org

1, 2, 4, 10, 18, 34, 64, 110, 188, 320, 524, 846, 1358, 2130, 3308, 5102, 7750, 11674, 17468, 25862, 38022, 55558, 80532, 116034, 166284, 236784, 335416, 472868, 663146, 925762, 1286920, 1780962, 2454792, 3370806, 4610656, 6284090, 8535868, 11554834, 15591564

Offset: 0

Seiichi Manyama, Oct 25 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Cf. A000122, A002448, A080054 ((theta_3(q^k)/theta_4(q^k))^(1/2)), A320098, A320967, A320992.

CoefficientList[Series[1/Product[EllipticTheta[4, 0, q^(2*k - 1)], {k, 1, 50}], {q, 0, 80}], q] (* G. C. Greubel, Oct 29 2018 *)

q='q+O('q^80); Vec(prod(k=1,50, eta(q^(2*k))^3/(eta(q^k)^2* eta(q^(4*k))) )) \\ G. C. Greubel, Oct 29 2018

a(n) = (-1)^n * A320098(n).
Expansion of Product_{k>0} eta(q^(2*k))^3 / (eta(q^k)^2*eta(q^(4*k))).
Expansion of Product_{k>0} 1/theta_4(q^(2*k-1)).

A261611 Expansion of Product_{k>=0} (1 + x^(4k+1))/(1 - x^(4k+1)).

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 6, 6, 6, 8, 12, 14, 14, 16, 22, 28, 30, 32, 40, 50, 56, 60, 70, 86, 98, 106, 120, 144, 166, 180, 200, 234, 270, 296, 324, 372, 428, 472, 514, 580, 664, 736, 800, 890, 1010, 1124, 1222, 1346, 1514, 1684, 1834, 2008, 2240, 2488, 2712, 2956

Offset: 0

Vaclav Kotesovec, Aug 26 2015

nonn

In general, if a > 0, b > 0, GCD(a,b) = 1 and g.f. = Product_{k>=0} (1 + x^(a*k+b))/(1 - x^(a*k+b)), then a(n) ~ Gamma(b/a) * a^(b/(2*a) - 1/2) * Pi^(b/a - 1) * exp(Pi*sqrt(n/a)) / (2^(2*b/a + 1) * n^(b/(2*a) + 1/2)).

Cf. A015128, A080054, A261610.

nmax = 60; CoefficientList[Series[Product[(1 + x^(4*k+1))/(1 - x^(4*k+1)), {k, 0, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(n)/2) * Gamma(1/4) / (2^(9/4) * Pi^(3/4) * n^(5/8)).

A115671 Number of partitions of n into parts not congruent to 0, 2, 12, 14, 16, 18, 20, 30 (mod 32).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 34, 44, 56, 72, 91, 114, 143, 178, 220, 272, 334, 408, 498, 605, 732, 884, 1064, 1276, 1528, 1824, 2171, 2580, 3058, 3616, 4269, 5028, 5910, 6936, 8124, 9498, 11088, 12922, 15034, 17468, 20264, 23472, 27154, 31369

Offset: 0

Michael Somos, Jan 29 2006

nonn

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

Andrews (1987) refers to this sequence as p(S, n) where S is the set in equation (1) on page 437.

			G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 11*x^8 + 15*x^9 + ...
a(5) = 4 since 5 = 4 + 1 = 3 + 1 + 1 = 1 + 1 + 1 + 1 + 1 in 4 ways.
a(6) = 6 since 6 = 5 + 1 = 4 + 1 + 1 = 3 + 3 = 3 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 + 1 in 6 ways.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..120 from Reinhard Zumkeller)
G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 32. MR0858826 (88b:11063).
G. E. Andrews, Unsolved Problems: Further Problems on Partitions, Amer. Math. Monthly 94 (1987), no. 5, 437-439.
Mircea Merca, The bisectional pentagonal number theorem, Journal of Number Theory, Volume 157, December 2015, Pages 223-232.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A080054, A187154, A208851, A208856, A218171.

a115671 = p [x | x <- [0..], (mod x 32) `notElem` [0,2,12,14,16,18,20,30]]
   where p _          0 = 1
         p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Mar 03 2012

a[ n_] := SeriesCoefficient[ (QPochhammer[ -q] / QPochhammer[ q] + 1) / 2, {q, 0, n}]; (* Michael Somos, Nov 09 2014 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^3 / QPochhammer[ q]^2 / QPochhammer[ q^4] + 1) / 2, {q, 0, n}]; (* Michael Somos, Nov 09 2014 *)

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

Expansion of (f(q) / f(-q) + 1) / 2 in powers of q where f() is a Ramanujan theta function.
Expansion of f(q^6, q^10) / f(-q, -q^3) = (f(q^22, q^26) - q^2 * f(q^10, q^38)) / f(-q, -q^2) in powers of x where f() is Ramanujan's two-variable theta function.
Euler transform of period 32 sequence [ 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, ...].
Given g.f. A(x), then B(x) = (2*A(x) - 1)^2 = g.f. A007096 satisfies 0 = f(B(x), B(x^2)) where f(u, v) = 1 + u^2 - 2 * u * v^2.
G.f. (1 + sqrt( theta_3(x) / theta_4(x))) / 2 = (Sum_{k} x^(8*k^2 - 2*k)) / (Sum_{k} (-x)^(2*k^2 - k)) = (Sum_{k} x^(24*n^2 + 2*n) - x^(24*n^2 + 14*n + 2)) / (Product_{k>0} 1 - x^k).
2 * a(n) = A080054(n) unless n = 0. a(2*n + 2) = A208851(n). a(2*n + 1) = A187154(n). a(n + 1) = A208856(n).

A080015 Expansion of theta_3(q) / theta_3(q^2) in powers of q.

Original entry on oeis.org

1, 2, -2, -4, 6, 8, -12, -16, 22, 30, -40, -52, 68, 88, -112, -144, 182, 228, -286, -356, 440, 544, -668, -816, 996, 1210, -1464, -1768, 2128, 2552, -3056, -3648, 4342, 5160, -6116, -7232, 8538, 10056, -11820, -13872, 16248, 18996, -22176, -25844, 30068

Offset: 0

Michael Somos, Jan 20 2003

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

			G.f. = 1 + 2*q - 2*q^2 - 4*q^3 + 6*q^4 + 8*q^5 - 12*q^6 - 16*q^7 + 22*q^8 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 214 Entry 24(ii).

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. A080054, A210030.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] / EllipticTheta[ 3, 0, q^2], {q, 0, n}]; (* Michael Somos, Apr 24 2015 *)

{a(n) = my(A, m); if( n<0, 0, m=1; A = 1 + 2 * x + O(x^2); while( m

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^8 + A) / eta(x + A))^2 * (eta(x^2 + A) / eta(x^4 + A))^7, n))};

Expansion of phi(q) / phi(q^2) in powers of q where phi() is a Ramanujan theta function.
Expansion of eta(q^2)^7 * eta(q^8)^2 / (eta(q)^2 * eta(q^4)^7) in powers of q.
Euler transform of period 8 sequence [ 2, -5, 2, 2, 2, -5, 2, 0, ...].
G.f.: A(x)/B(x), where A(x) = Sum_{m = -infinity..infinity} x^(m^2) and B(x) = Sum_{m = -infinity..infinity} x^(2*m^2). - Vladeta Jovovic, Mar 22 2005
Expansion of phi(x) / phi(x^2) where phi() is a Ramanujan theta function.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 1 + (1 - u*v)^2 - v^2. - Michael Somos, Jan 31 2006
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^4 - v^4 + 8 * u*v - 6 * u*v * (u^2 + v^2) + 4 * (u*v)^3. - Michael Somos, Jan 31 2006
Expansion of sqrt(m) in powers of q where m is the multiplier for the second degree modular equation.
G.f.: Prod_{k>0} ((1 - x^(8*k - 2)) * (1 - x^(8*k - 6)))^5 / ((1 - x^(8*k - 1)) * (1 - x^(8*k - 3)) * (1 - x^(8*k - 4)) * (1 - x^(8*k - 5)) * (1 - x^(8*k - 7)))^2.
a(n) = (-1)^n * A210030(n). a(n) = (-1)^[n/2] * A080054(n).

A108494 Expansion of f(-q) / f(q) in powers of q where f() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 2, -4, 6, -8, 12, -16, 22, -30, 40, -52, 68, -88, 112, -144, 182, -228, 286, -356, 440, -544, 668, -816, 996, -1210, 1464, -1768, 2128, -2552, 3056, -3648, 4342, -5160, 6116, -7232, 8538, -10056, 11820, -13872, 16248, -18996, 22176, -25844, 30068, -34936, 40528

Offset: 0

Michael Somos, Jun 06 2005

sign

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

			1 - 2*q + 2*q^2 - 4*q^3 + 6*q^4 - 8*q^5 + 12*q^6 - 16*q^7 + 22*q^8 - ...

T. Miwa, Integrable Lattice Models and Branching Coefficients, Proceedings of the International Congress of Mathematicians, Vol. 1, (Berkeley, Calif., 1986), 862-870, Amer. Math. Soc., Providence, RI, 1987. MR0934288 (89h:82051)

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. A080054.

CoefficientList[QPochhammer[q]/QPochhammer[-q] + O[q]^50, q] (* Jean-François Alcover, Nov 05 2015 *)

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

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

Expansion of (1 - k^2)^(1/8) = k'^(1/4) in powers of q = exp(-Pi K'/K).
Expansion of (theta_4(q) / theta_3(q))^(1/2) = (phi(-q) / phi(q))^(1/2) = chi(-q) / chi(q) = psi(-q) / psi(q) = f(-q) / f(q) where phi(), chi(), psi(), f() are Ramanujan theta functions.
Expansion of eta(q)^2 eta(q^4) / eta(q^2)^3 in powers of q.
Euler transform of period 4 sequence [ -2, 1, -2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = -2*u^2 + v^4 + u^4*v^4.
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^4 + 2*u*v -2*u^3*v^3 - v^4.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = -2*u + v*w^2 + u^2*v*w^2.
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*u2*u6 - 2*u1*u3 + u2*u3^2*u6.
G.f. A(x) satisfies 0 = f(A(x), A(x^5)) where f(u, v) = 4 * u*v *(1 - u^4) * (1 + v^4) - (v^2 - u^2) * (u + v)^4. - Michael Somos, Sep 11 2006
G.f. A(x) satisfies 0 = f(A(x), A(x^7)) where f(u, v) = (1 - u^8) * (1 - v^8) - (1 - u*v)^8. - Michael Somos, Jan 01 2006
G.f.: Product_{k>0} (1 - x^(2*k - 1)) / (1 + x^(2*k - 1)).
a(n) = (-1)^n * A080054(n). Convolution inverse of A080054.
Empirical: sum(exp(-Pi)^(n-1)*(-1)^(n+1)*a(n),n=1..infinity) = 2^(1/8). - Simon Plouffe, Feb 20 2011
Empirical: sum(exp(-Pi)^(n-1)*a(n),n=1..infinity) = 2^(7/8)/2. - Simon Plouffe, Feb 20 2011
a(n) ~ (-1)^n * exp(Pi*sqrt(n/2)) / (2^(9/4) * n^(3/4)). - Vaclav Kotesovec, Oct 23 2017
G.f.: exp(-2*Sum_{k>=1} sigma(2*k - 1)*x^(2*k-1)/(2*k - 1)). - Ilya Gutkovskiy, Apr 19 2019

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A007096 Expansion of theta_3 / theta_4.

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A098151 Number of partitions of 2*n with no part divisible by 3 and all odd parts occurring with even multiplicities.

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A261610 Expansion of Product_{k>=0} (1 + x^(3*k+1))/(1 - x^(3*k+1)).

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A320968 Expansion of (Product_{k>0} theta_3(q^k)/theta_4(q^k))^(1/2), where theta_3() and theta_4() are the Jacobi theta functions.

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A261611 Expansion of Product_{k>=0} (1 + x^(4*k+1))/(1 - x^(4*k+1)).

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A115671 Number of partitions of n into parts not congruent to 0, 2, 12, 14, 16, 18, 20, 30 (mod 32).

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