A080054 -id:A080054 - cp's OEIS frontend

A300413 Expansion of Product_{k>=1} (1 + x^prime(k))/(1 - x^prime(k)).

1, 0, 2, 2, 2, 6, 4, 10, 10, 14, 20, 22, 32, 38, 48, 60, 74, 90, 112, 134, 164, 196, 236, 282, 336, 398, 472, 554, 652, 766, 890, 1046, 1206, 1408, 1624, 1876, 2168, 2486, 2860, 3276, 3744, 4282, 4878, 5554, 6316, 7160, 8124, 9186, 10388, 11722, 13216, 14876, 16732, 18794, 21084, 23636

Offset: 0

Ilya Gutkovskiy, Mar 05 2018

nonn

Convolution of the sequences A000586 and A000607.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Index entries for related partition-counting sequences

Cf. A000040, A000586, A000607, A015128, A080054, A103265, A280263, A280366, A300414, A300415.

nmax = 55; CoefficientList[Series[Product[(1 + x^Prime[k])/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A000040(k))/(1 - x^A000040(k)).

log(a(n)) ~ Pi*sqrt(2*n/log(n/3)) * (1/3 + 2*sqrt(log(n/3) / log(2*n/3)) / 3). - Vaclav Kotesovec, Jan 12 2021

A361008 G.f.: Product_{k >= 0} ((1 + x^(2k+1)) / (1 - x^(2k+1)))^k.

Original entry on oeis.org

1, 0, 0, 2, 0, 4, 2, 6, 8, 10, 20, 18, 42, 40, 78, 92, 140, 192, 258, 382, 480, 728, 902, 1334, 1698, 2404, 3148, 4292, 5742, 7608, 10304, 13430, 18192, 23592, 31720, 41144, 54766, 71188, 93762, 122156, 159420, 207820, 269380, 350726, 452434, 587520, 755446

Offset: 0

Vaclav Kotesovec, Apr 20 2023

nonn

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

Cf. A080054, A291697, A263149, A263150.

Cf. A156616, A270919.

Table[SeriesCoefficient[Product[((1 + x^(2*k + 1))/(1 - x^(2*k + 1)))^k, {k, 0, n}], {x, 0, n}], {n, 0, 50}]

a(n) ~ sqrt(A/(3*Pi)) * (7*zeta(3))^(11/72) * exp(3*(7*zeta(3))^(1/3) * n^(2/3)/4 - Pi^2 * n^(1/3)/(8*(7*zeta(3))^(1/3)) - 1/24 - Pi^4/(1344*zeta(3))) / (2^(3/4) * n^(47/72)), where A = A074962 is the Glaisher-Kinkelin constant.

A100684 Number of partitions of 2n free of multiples of 8 such that 4 occurs at most once. All odd parts occur with even multiplicities. There is no restriction on the other even parts.

Original entry on oeis.org

1, 2, 4, 8, 12, 20, 32, 48, 72, 106, 152, 216, 305, 422, 580, 792, 1068, 1432, 1908, 2520, 3313, 4332, 5628, 7280, 9373, 12008, 15324, 19480, 24661, 31112, 39120, 49016, 61229, 76260, 94692, 117264, 144834, 178412, 219244, 268784, 328746

Offset: 0

Noureddine Chair, Jan 27 2005

nonn

			G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 12*x^4 + 20*x^5 + 32*x^6 + 48*x^7 + 72*x^8 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.

Cf. A080054.

nmax = 40; CoefficientList[Series[(1-x^4)*Product[(1+x^(2*k))/(1-x^(2*k-1))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 06 2015 *)

{a(n) = my(A); if( n<0, 0, A = x*O(x^n); polcoeff( (1 - x^4) * eta(x^4 + A) * eta(x^2 + A) / eta(x + A)^2, n))}; /* Michael Somos, Feb 10 2005 */

G.f.: (1-x^4)*Product((1+x^(2*i))/(1-x^(2*i-1))^2, i=1..infinity). [Vladeta Jovovic]
Expansion of (1 - q^4) * q^(-1/6) * eta(q^4) * eta(q^2) / eta(q)^2 in powers of q.
G.f.: (1-x^4) * Prod_{k>0} (1 + x^(2*k)) * (1 + x^k)^2. - Michael Somos, Feb 10 2005
a(n) ~ 5^(3/4) * Pi * exp(Pi*sqrt(5*n/6)) / (2^(11/4) * 3^(3/4) * n^(5/4)). - Vaclav Kotesovec, Sep 06 2015

Corrected by Vladeta Jovovic, Feb 01 2005

Typo in PARI program fixed by Vaclav Kotesovec, Sep 06 2015

A193863 Expansion of Product_{n>=0} (1 + q(-q^2)^n) / (1 - q(-q^2)^n).

Original entry on oeis.org

1, 2, 2, 0, -2, 0, 4, 4, -2, -6, 0, 8, 4, -8, -8, 8, 14, -4, -18, 0, 24, 8, -28, -20, 28, 34, -24, -48, 16, 64, 0, -76, -18, 88, 44, -96, -78, 96, 116, -88, -160, 68, 208, -32, -252, -16, 296, 84, -332, -170, 354, 272, -360, -392, 344, 528, -296, -672, 216, 824, -96, -976, -72, 1116, 286, -1240, -552, 1336, 876, -1384

Offset: 0

Joerg Arndt, Aug 07 2011

sign

Expansion of E(-q^2, +q) for E(q,x) = Product_{n>=0} ( 1 + x*q^n ) / ( 1 - x*q^n ).

Replacing q by -q in the g.f. gives the inverse of the g.f., whose expansion is obtained by negating every second term.

			1 + 2*x + 2*x^2 - 2*x^4 + 4*x^6 + 4*x^7 - 2*x^8 - 6*x^9 + 8*x^11 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
J. Arndt, Regularized q-exponential and q-trigonometric series of the first kind

Cf. A015128 E(+q,+q), A002448 E(+q,-q), A000122 E(-q,+q), A004402 E(-q,-q), A080054 E(+q^2,+q), A108494 E(+q^2,-q), A300574, A300575.

N=66; q='q+O('q^N); /* that many terms */
gf = prod(n=0, N, (1+q*(-q^2)^n)/(1-q*(-q^2)^n) );
Vec(gf) /* show terms */
/* Alternative computation of the g.f. using a product form */
V=[0,-2, 1, 2, 0, -2, -1, 2]; /* note vectors are one-based */
gf=prod(n=0, N, (1-q^n)^(V[n%8+1]) );

{a(n) = local(A); if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k + x * O(x^n) )^[ 0, -2, 1, 2, 0, -2, -1, 2][k%8 + 1]), n))} /* Michael Somos, Feb 26 2012 */

Euler transform of period 8 sequence [ 2, -1, -2, 0, 2, 1, -2, 0, ...]. - Michael Somos, Feb 26 2012
G.f.: prod(n>=0, (1+q*(-q^2)^n)/(1-q*(-q^2)^n) ).
G.f.: sum(n>=0, prod(k=0..n-1, 1+(-q^2)^k )/prod(k=1..n, 1-(-q^2)^k ) * q^n ).
G.f.: sum(n>=0, prod(k=0..n-1, 1+(-q^2)^k)/( prod(k=1..n, 1-(-q^2)^k) * prod(k=0..n-1, 1-q*(-q^2)^k ) ) * q^n * (-q^2)^(n*(n-1)/2) ).
Convolution of A300574 and A300575. - Seiichi Manyama, Nov 22 2019

A274352 Convolution of A015723 and A000700.

Original entry on oeis.org

0, 1, 2, 4, 7, 10, 18, 26, 36, 53, 76, 104, 140, 190, 252, 336, 437, 564, 732, 936, 1186, 1504, 1894, 2366, 2950, 3659, 4520, 5564, 6822, 8330, 10152, 12326, 14906, 17996, 21662, 25996, 31135, 37190, 44314, 52704, 62532, 74036, 87504, 103212, 121496, 142798

Offset: 0

R. J. Mathar, Jun 18 2016

nonn

Also the convolution of A080054 and A048272.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
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, corollary 3.4

Cf. A015723, A000700, A080054, A048272.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
     [0, 2, -1, 2][1+irem(d, 4)], d=divisors(j))*b(n-j), j=1..n)/n)
    end:
g:= proc(n) option remember; add((-1)^(d+1), d=divisors(n)) end:
a:= n-> add(b(j)*g(n-j), j=0..n):
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 18 2016

q[n_, k_] := q[n, k] = If[n < k || k < 1, 0, If[n == 1, 1, q[n - k, k] + q[n - k, k - 1]]]; Table[Sum[SeriesCoefficient[Product[1 + x^j, {j, 1, k, 2}], {x, 0, k}] Sum[i q[#, i], {i, 1, Floor[(Sqrt[8 # + 1] - 1)/2]}] &[n - k], {k, 0, n}], {n, 0, 45}] (* Michael De Vlieger, Jun 18 2016, after Vaclav Kotesovec at A015723 and Vladimir Reshetnikov at A000700 *)

a(n) = Sum_{k=0..n} A015723(k)*A000700(n-k).

a(n) ~ log(2) * exp(Pi*sqrt(n/2)) / (Pi * 2^(3/4) * n^(1/4)). - Vaclav Kotesovec, Sep 14 2021

A294592 a(n) = [x^n] (theta_3(x)/theta_4(x))^n, where theta_() is the Jacobi theta function.

Original entry on oeis.org

1, 4, 32, 304, 3072, 32024, 340352, 3666016, 39878656, 437091892, 4819567552, 53401892240, 594093969408, 6631726263608, 74242911364864, 833237193123104, 9371924860764160, 105614054423502408, 1192210691317862048, 13478559927485340144, 152589996020498655232, 1729590806617202662528

Offset: 0

Ilya Gutkovskiy, Nov 03 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..500
Eric Weisstein's MathWorld, Jacobi Theta Functions
Wikipedia, Theta function

Cf. A007096, A014969, A014970, A014972, A066535, A080054, A103261, A270919, A289522, A291697.

S:= series((JacobiTheta3(0,x)/JacobiTheta4(0,x))^n,x,51):
seq(coeff(S,x,n),n=0..50); # Robert Israel, Nov 03 2017

Table[SeriesCoefficient[(EllipticTheta[3, 0, x]/EllipticTheta[4, 0, x])^n, {x, 0, n}], {n, 0, 21}]
Table[SeriesCoefficient[Product[((1 + x^(2 k + 1))/(1 - x^(2 k + 1)))^(2 n), {k, 0, n}], {x, 0, n}], {n, 0, 21}]
Table[SeriesCoefficient[(QPochhammer[-x, x^2]/QPochhammer[x, x^2])^(2 n), {x, 0, n}], {n, 0, 21}]
(* Calculation of constant d: *) 1/r /. FindRoot[{s == EllipticTheta[3, 0, r*s]/EllipticTheta[4, 0, r*s], EllipticTheta[4, 0, r*s] + r*s*Derivative[0, 0, 1][EllipticTheta][4, 0, r*s] == r*Derivative[0, 0, 1][EllipticTheta][3, 0, r*s]}, {r, 1/10}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Jan 17 2024 *)

a(n) = [x^n] Product_{k>=0} ((1 + x^(2*k+1))/(1 - x^(2*k+1)))^(2*n).
From Vaclav Kotesovec, Nov 05 2017: (Start)
a(n) ~ c * d^n / sqrt(n), where
d = 11.61255065799699699891360038489317237925475956178123836149123386457... and
c = 0.34456510029264878768512693687607064416428117641473856418257649837... (End)

A300414 Expansion of Product_{k>=2} (1 + x^Fibonacci(k))/(1 - x^Fibonacci(k)).

Original entry on oeis.org

1, 2, 4, 8, 12, 20, 30, 42, 62, 84, 114, 154, 198, 260, 332, 418, 530, 654, 810, 994, 1202, 1462, 1752, 2094, 2500, 2948, 3486, 4092, 4776, 5582, 6468, 7490, 8650, 9928, 11406, 13036, 14862, 16934, 19196, 21758, 24592, 27706, 31216, 35038, 39284, 43990, 49100, 54798, 61008, 67798

Offset: 0

Ilya Gutkovskiy, Mar 05 2018

nonn

Convolution of the sequences A000119 and A003107.

Index entries for related partition-counting sequences

Cf. A000045, A000119, A003107, A015128, A080054, A103265, A280263, A280366, A300413, A300415.

nmax = 49; CoefficientList[Series[Product[(1 + x^Fibonacci[k])/(1 - x^Fibonacci[k]), {k, 2, 20}], {x, 0, nmax}], x]

G.f.: Product_{k>=2} (1 + x^A000045(k))/(1 - x^A000045(k)).

A304627 a(n) = [x^n] Product_{k>=1} (1 + x^k)(1 - x^(nk))/((1 - x^k)(1 + x^(nk))).

Original entry on oeis.org

1, 0, 2, 6, 12, 22, 38, 62, 98, 152, 230, 342, 502, 726, 1038, 1470, 2060, 2862, 3946, 5398, 7334, 9902, 13286, 17726, 23526, 31064, 40822, 53406, 69566, 90246, 116622, 150142, 192610, 246254, 313806, 398638, 504884, 637590, 802934, 1008446, 1263270, 1578526, 1967694, 2447062

Offset: 0

Ilya Gutkovskiy, May 15 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to partitions

Cf. A000065, A015128, A052839, A080054, A098151, A103258, A111133, A138526, A262968.

Table[SeriesCoefficient[Product[(1 + x^k) (1 - x^(n k))/((1 - x^k) (1 + x^(n k))) , {k, 1, n}], {x, 0, n}], {n, 0, 43}]
Table[SeriesCoefficient[Product[(1 + x^k)/(1 - x^k), {k, 1, n - 1}], {x, 0, n}], {n, 0, 43}]
Join[{1}, Table[SeriesCoefficient[EllipticTheta[4, 0, x^n]/EllipticTheta[4, 0, x], {x, 0, n}], {n, 43}]]
nmax = 43; CoefficientList[Series[1/EllipticTheta[4, 0, x] - 2 x/(1 - x), {x, 0, nmax}], x]

G.f.: 1/theta_4(x) - 2*x/(1 - x), where theta_4() is the Jacobi theta function.

a(n) ~ exp(Pi*sqrt(n)) / (8*n). - Vaclav Kotesovec, May 19 2018

A208150 Expansion of (1 + 2Product_{i>=1} (1 + x^(8i - 1)))/(-1 + Product_{i>=1} (1 - x^(8*i - 1))).

Original entry on oeis.org

-2, 3, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, -4, 3, 0, 0, 0, 0, 0, 0, -3, -3, 0, 0, 0, 0, 0, -4, 2, 3, 0, 0, 0, 0, 0, -3, -9, -3, 0, 0, 0, 0, -4, 1, 5, 3, 0, 0, 0, 0, 1, -15, -9, -3, 0, 0, 0, -4, 0, 6, 5, 3, 0, 0, 0, 5, -20, -21, -12, -3, 0, 0, -4, -1, 16

Offset: 0

Roger L. Bagula, Feb 23 2012

			-3/x^7 - 2 + 3*x -3*x^8 - 4*x^15 + 3*x^16 - 3*x^23 - 3*x^24 - 4*x^30 + ...

Cf. A207944, A080054, A142724.

Table[SeriesCoefficient[
  Series[(1 + 2*Product[1 + x^(8*i - 1), {i, 1, Infinity}])/(-1 +
      Product[(1 - x^(8*i - 1)), {i, 1, Infinity}]), {x, 0, 100}],
  n], {n, 0, 100}]

A216879 G.f. satisfies: A(x) = sqrt( theta_3(xA(x)) / theta_4(xA(x)) ).

Original entry on oeis.org

1, 2, 6, 24, 110, 540, 2772, 14704, 79974, 443594, 2499640, 14269320, 82346004, 479604748, 2815557264, 16643093712, 98974828886, 591742372068, 3554708076858, 21444913596408, 129870710693976, 789237890852160, 4811481299622276, 29417496447990096, 180337119342194820

Offset: 0

Paul D. Hanna, Sep 18 2012

nonn

Here theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2) and theta_4(x) = 1 + 2*Sum_{n>=1} (-x)^(n^2) are Jacobi theta functions.
The radius of convergence r of g.f. A(x) is given by
r = 0.15335406881552899483841215094726329935743212998703... with
A(r) = 2.14877235788136654366723937779352044712735012012453...
such that G(y) = y*G'(y) = A(r) at y = r*A(r) = 0.3295229840394455820300...
where G(x) = sqrt(theta_3(x)/theta_4(x)).
Conjectured to be the number of permutations of length n+1 avoiding the partially ordered pattern (POP) {5>1, 1>2, 1>4} of length 5. That is, conjectured to be the number of length n+1 permutations having no subsequences of length 5 in which the fifth element is larger than the first element, which in turn is larger than the second and fourth elements. - Sergey Kitaev, Dec 13 2020
The conjecture was disproven. The numbers are actually A366706, which matches the first 9 entries. - Christian Bean, Jul 22 2024

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 24*x^3 + 110*x^4 + 540*x^5 + 2772*x^6 +...
such that, by definition, the g.f. satisfies:
A(x) = sqrt( (1 + 2*Sum_{n>=1} (x*A(x))^(n^2) ) / (1 + 2*Sum_{n>=1} (-x*A(x))^(n^2) ) ).

Christian Bean, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, Permutations avoiding bipartite partially ordered patterns have a regular insertion encoding, The Electronic Journal of Combinatorics, Volume 31, Issue 3 (2024); arXiv preprint, arXiv:2312.07716 [math.CO], 2023.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.

Cf. A080054.

InverseSeries[x Sqrt[EllipticTheta[4, 0, x]/EllipticTheta[3, 0, x]] + O[x]^26] // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Oct 01 2019 *)
(* Calculation of constants {d,c}: *) {1/r, s*Sqrt[EllipticTheta[3, 0, r*s] / (Pi*(6*EllipticTheta[3, 0, r*s] - r*s*(4*Derivative[0, 0, 1][EllipticTheta][3, 0, r*s] - r*s*Derivative[0, 0, 2][EllipticTheta][3, 0, r*s] + r*s^3*Derivative[0, 0, 2][EllipticTheta][4, 0, r*s])))]} /. FindRoot[{EllipticTheta[3, 0, r*s]/EllipticTheta[4, 0, r*s] == s^2, (r*(Derivative[0, 0, 1][EllipticTheta][3, 0, r*s] - s^2*Derivative[0, 0, 1][EllipticTheta][4, 0, r*s])) / (2*s*EllipticTheta[4, 0, r*s]) == 1}, {r, 1/6}, {s, 3/2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Nov 16 2023 *)

{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=sqrt((1+2*sum(m=1,sqrtint(n)+1,(x*A)^(m^2)))/(1+2*sum(m=1,sqrtint(n)+1,(-x*A)^(m^2)))));polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=eta(-x*A)^2*eta(x^4*A^4)/eta(x^2*A^2)^3);polcoeff(A,n)}

{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp(2*sum(n=1,n,sigma(2*n-1)*(x*A)^(2*n-1)/(2*n-1))));polcoeff(A,n)}

{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1/prod(m=1,n,(1+(x*A)^(2*m))*(1-(x*A)^(2*m-1))^2));polcoeff(A,n)}

{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=prod(m=1,n,(1+(x*A)^(2*m-1))*(1+(x*A)^m)));polcoeff(A,n)}

{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=prod(m=1,n,(1+(x*A)^(2*m-1))/(1-(x*A)^(2*m-1))));polcoeff(A,n)}

{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=prod(m=1,n,(1-(x*A)^(4*m-2))/((1-(x*A)^(4*m-1))*(1-(x*A)^(4*m-3)))^2));polcoeff(A,n)}

G.f. satisfies the identities:
(1) A(x) = 1 / A(-x*A(x)^2).
(2) A(x) = eta(-x*A(x))^2 * eta(x^4*A(x)^4) / eta(x^2*A(x)^2)^3.
(3) A(x) = exp( 2*Sum_{n>=1} sigma(2*n-1) * (x*A(x))^(2*n-1) / (2*n-1) ).
(4) A(x) = 1 / Product_{n>=1} (1 + (x*A(x))^(2*n)) * (1 - (x*A(x))^(2*n-1))^2.
(5) A(x) = Product_{n>=1} (1 + (x*A(x))^(2*n-1)) * (1 + (x*A(x))^n).
(6) A(x) = Product_{n>=1} (1 + (x*A(x))^(2*n-1)) / (1 - (x*A(x))^(2*n-1)).
(7) A(x) = Product_{n>=1} (1 - (x*A(x))^(4*n-2)) / ((1 - (x*A(x))^(4*n-1))*(1 - (x*A(x))^(4*n-3)))^2.
(8) A(x) = 1/(1 - 2*q/(1+q - q^2*(1-q^2)/(1+q^3 - q^3*(1-q^4)/(1+q^5 - q^4*(1-q^6)/(1+q^7 - ...))))), a continued fraction, where q = x*A(x).
(9) A(x) = (1/x)*Series_Reversion( x*sqrt(theta_4(x)/theta_3(x)) ).
(10) A(x/G(x)) = G(x) where G(x) = sqrt(theta_3(x)/theta_4(x)) is the g.f. of A080054.
Special value: A(exp(-Pi)/2^(1/8)) = 2^(1/8).
a(n) = [x^n] ( theta_3(x) / theta_4(x) )^((n+1)/2) / (n+1).
a(n) ~ c * d^n / n^(3/2), where d = 6.52085730573545526010335599231748172235904166255252115709479430152403... and c = 0.6370998492207183978277090515469899143891211207560886906399176320450... - Vaclav Kotesovec, Nov 16 2023

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A300413 Expansion of Product_{k>=1} (1 + x^prime(k))/(1 - x^prime(k)).

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A361008 G.f.: Product_{k >= 0} ((1 + x^(2*k+1)) / (1 - x^(2*k+1)))^k.

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A100684 Number of partitions of 2n free of multiples of 8 such that 4 occurs at most once. All odd parts occur with even multiplicities. There is no restriction on the other even parts.

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A193863 Expansion of Product_{n>=0} (1 + q*(-q^2)^n) / (1 - q*(-q^2)^n).

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Formula

A274352 Convolution of A015723 and A000700.

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Maple

Mathematica

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A294592 a(n) = [x^n] (theta_3(x)/theta_4(x))^n, where theta_() is the Jacobi theta function.

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Maple

Mathematica

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A300414 Expansion of Product_{k>=2} (1 + x^Fibonacci(k))/(1 - x^Fibonacci(k)).

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Mathematica

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A304627 a(n) = [x^n] Product_{k>=1} (1 + x^k)*(1 - x^(n*k))/((1 - x^k)*(1 + x^(n*k))).

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A361008 G.f.: Product_{k >= 0} ((1 + x^(2k+1)) / (1 - x^(2k+1)))^k.

A193863 Expansion of Product_{n>=0} (1 + q(-q^2)^n) / (1 - q(-q^2)^n).

A304627 a(n) = [x^n] Product_{k>=1} (1 + x^k)(1 - x^(nk))/((1 - x^k)(1 + x^(nk))).

A208150 Expansion of (1 + 2Product_{i>=1} (1 + x^(8i - 1)))/(-1 + Product_{i>=1} (1 - x^(8*i - 1))).

A216879 G.f. satisfies: A(x) = sqrt( theta_3(xA(x)) / theta_4(xA(x)) ).