A003105 -id:A003105 - cp's OEIS frontend

A261771 Expansion of Product_{k>=1} (1 + x^k) / (1 + x^(8*k)).

1, 1, 1, 2, 2, 3, 4, 5, 5, 7, 9, 10, 13, 15, 18, 22, 26, 30, 36, 42, 49, 58, 67, 77, 89, 103, 118, 136, 156, 177, 203, 231, 263, 299, 338, 383, 433, 489, 550, 620, 696, 781, 877, 981, 1097, 1227, 1369, 1526, 1702, 1893, 2104, 2339, 2595, 2877, 3189, 3530

Offset: 0

Vaclav Kotesovec, Aug 31 2015

nonn

a(n) is the number of partitions of n into distinct parts where no part is a multiple of 8.

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

Cf. A261735.
Cf. A000700 (m=2), A003105 (m=3), A070048 (m=4), A096938 (m=5), A261770 (m=6), A097793 (m=7), A112193 (m=9), A261772 (m=10).
Column k=8 of A290307.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
       [0, 1, 0, 1, 0, 1, 0, 1, -1, 1, 0, 1, 0, 1, 0, 1]
       [1+irem(d, 16)], d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 31 2015

nmax = 50; CoefficientList[Series[Product[(1 + x^k) / (1 + x^(8*k)), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(7*n/6)/2) * 7^(1/4) / (4 * 6^(1/4) * n^(3/4)) * (1 - (3*sqrt(3)/ (2*Pi*sqrt(14)) + 7*Pi*sqrt(7)/(96*sqrt(6))) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017

G.f.: Product_{k>=1} (1 - x^(16*k-8))/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 07 2017

A261772 Expansion of Product_{k>=1} (1 + x^k) / (1 + x^(10*k)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 16, 20, 24, 28, 33, 40, 46, 54, 64, 74, 86, 100, 115, 133, 154, 176, 202, 231, 263, 300, 342, 388, 440, 499, 563, 636, 718, 808, 909, 1022, 1146, 1284, 1439, 1608, 1797, 2006, 2236, 2490, 2772, 3081, 3422, 3800, 4212

Offset: 0

Vaclav Kotesovec, Aug 31 2015

nonn

a(n) is the number of partitions of n into distinct parts where no part is a multiple of 10.

In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^k)/(1 + x^(m*k)), then a(n) ~ exp(Pi*sqrt((m-1)*n/(3*m))) * (m-1)^(1/4) / (2^(3/2) * 3^(1/4) * m^(1/4) * n^(3/4)) * (1 - (3*sqrt(3*m)/(8*Pi*sqrt(m-1)) + (m-1)^(3/2)*Pi/(48*sqrt(3*m))) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017

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

Cf. A145707.
Cf. A000700 (m=2), A003105 (m=3), A070048 (m=4), A096938 (m=5), A261770 (m=6), A097793 (m=7), A261771 (m=8), A112193 (m=9).
Column k=10 of A290307.

b:= proc(n, i) option remember;  local r;
      `if`(2*n>i*(i+1)-(j-> 10*j*(j+1))(iquo(i, 10, 'r')), 0,
      `if`(n=0, 1, b(n, i-1)+`if`(i>n or r=0, 0, b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 31 2015

nmax = 50; CoefficientList[Series[Product[(1 + x^k) / (1 + x^(10*k)), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(3*n/10)) * 3^(1/4) / (2^(7/4) * 5^(1/4) * n^(3/4)) * (1 - (sqrt(15)/(4*Pi*sqrt(2)) + 3*Pi*sqrt(3)/(16*sqrt(10))) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017

G.f.: Product_{k>=1} (1 - x^(20*k-10))/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 07 2017

A003113 Coefficients in expansion of permanent of infinite tridiagonal matrix shown below.

Original entry on oeis.org

2, 1, 2, 2, 3, 3, 5, 5, 7, 8, 10, 11, 15, 16, 20, 23, 28, 31, 38, 42, 51, 57, 67, 75, 89, 99, 115, 129, 149, 166, 192, 213, 244, 272, 309, 344, 391, 433, 489, 543, 611, 676, 760, 839, 939, 1038, 1157, 1276, 1422, 1565, 1738, 1913, 2119, 2328, 2576, 2826, 3120

Offset: 0

N. J. A. Sloane, Herman P. Robinson

nonn

1 0 0 0 0 0 ...
1 x 0 0 0 0 0 ...
x 1 x^2 0 0 0 ...
0 x^2 1 x^3 0 0 ...
0 0 x^3 1 x^4 0 0 0 ...
...................

D. H. Lehmer, Course on History of Mathematics, Univ. Calif. Berkeley, 1973.
H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Herman P. Robinson, Letter to N. J. A. Sloane, Jan 1974.

Cf. A119469, A003114, A003106.

The generalized Rogers-Ramanujan series G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] are A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595. The present sequence, which is G[1]+G[2], plays the role of G[0].

nmax = 60; CoefficientList[1 + Series[Sum[x^(j*(j-1))/Product[1 - x^i, {i, 1, j}], {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 02 2016 *)

G.f.: 1 + sum(i>=1, x^(i*(i-1))/prod(j=1..i, 1-x^j)) - Jon Perry, Jul 04 2004
a(n) = A003114(n)+A003106(n). So this is the sum of the two famous Rogers-Ramanujan series. - Vladeta Jovovic, Jul 17 2004
G.f.: sum(n>=0,(q^(n^2)*(1+q^n)) / prod(k=1..n,1-q^k)). [Joerg Arndt, Oct 08 2012]
a(n) ~ (9+4*sqrt(5))^(1/4) * exp(2*Pi*sqrt(n/15)) / (2*3^(1/4)*sqrt(5)*n^(3/4)). - Vaclav Kotesovec, Jan 02 2016

More terms from Vladeta Jovovic, Aug 30 2001

A132463 Number of partitions of n into distinct parts congruent to 0 or 1 modulo 3.

Original entry on oeis.org

1, 1, 0, 1, 2, 1, 1, 3, 2, 2, 5, 4, 3, 7, 7, 5, 10, 11, 8, 14, 17, 13, 20, 25, 19, 27, 36, 29, 37, 50, 43, 51, 69, 61, 69, 94, 86, 93, 126, 120, 125, 167, 164, 167, 220, 222, 222, 287, 297, 294, 373, 393, 386, 481, 516, 505, 617, 672, 657, 788, 868, 850, 1002, 1114, 1094

Offset: 0

Reinhard Zumkeller, Aug 22 2007

nonn

			a(7)=3 because we have 7, 61 and 43.

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from Reinhard Zumkeller)

Cf. A032766, A035360, A003105, A132462.

g:=product((1+x^(3*k))*(1+x^(3*k-2)),k=1..30): gser:=series(g,x=0,100): seq(coeff(gser,x,n),n=0..65); # Emeric Deutsch, Aug 26 2007

nmax = 100; CoefficientList[Series[Product[((1+x^(3*k))*(1+x^(3*k-2))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 24 2015 *)

G.f.: Product(k>=1, (1+x^(3*k))*(1+x^(3*k-2)) ). - Emeric Deutsch, Aug 26 2007

a(n) ~ exp(Pi*sqrt(2*n)/3) / (2^(19/12) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Aug 24 2015

Prepended a(0) = 1, Joerg Arndt, Feb 22 2015

A132462 Number of partitions of n into distinct parts congruent to 0 or 2 modulo 3.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 1, 1, 3, 2, 2, 5, 2, 4, 7, 4, 7, 10, 6, 11, 14, 9, 17, 19, 14, 25, 26, 21, 36, 35, 31, 50, 47, 45, 69, 63, 64, 93, 84, 89, 125, 111, 124, 165, 147, 169, 216, 194, 227, 281, 254, 303, 363, 332, 400, 466, 432, 523, 595, 559, 680, 756, 721, 876, 956, 926, 1121

Offset: 0

Reinhard Zumkeller, Aug 22 2007

nonn

			a(8)=3 because we have 8, 6+2 and 5+3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..200

Cf. A007494, A035361, A003105, A132463.

g:=product((1+x^(3*k))*(1+x^(3*k-1)),k=1..30): gser:=series(g,x=0,100): seq(coeff(gser,x,n),n=0..70); # Emeric Deutsch, Aug 30 2007

nmax = 40; CoefficientList[Series[Product[((1+x^(3*k))*(1+x^(3*k-1))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 24 2015 *)

G.f.: Product_{k>=1} (1+x^(3*k))*(1+x^(3*k-1)). - Emeric Deutsch, Aug 30 2007

a(n) ~ exp(Pi*sqrt(2*n)/3) / (2^(23/12) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Aug 24 2015

a(0)=1 prepended by Vaclav Kotesovec, Aug 24 2015

A096981 Number of partitions of n into parts congruent to {0, 1, 3, 5} mod 6.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 6, 7, 10, 12, 15, 21, 25, 30, 39, 46, 56, 72, 85, 101, 125, 147, 175, 215, 252, 296, 356, 415, 487, 582, 676, 786, 927, 1072, 1244, 1460, 1682, 1939, 2255, 2588, 2976, 3446, 3942, 4510, 5189, 5916, 6751, 7739, 8797, 9999, 11406, 12927, 14657

Offset: 0

Noureddine Chair, Aug 19 2004

nonn

Also, number of partitions of n in which the distinct parts are prime to 3 and the unrestricted parts are multiples of 3.

The inverted graded parafermionic partition function. This g.f. is a generalization of A003105, A006950 and A096938

			a(11) = 15 because we can write 11 = 10+1 = 8+2+1 = 7+4 = 5+4+2 (parts do not contain multiple of 3) = 9+2 = 8+3 = 7+3+1 = 6+5 = 6+4+1 = 6+3+2 = 5+3+3 = 5+3+2+1 = 4+3+3+1 = 3+3+3+2.
1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 6*x^7 + 7*x^8 + 10*x^9 + ...
q^-5 + q^19 + q^43 + 2*q^67 + 2*q^91 + 3*q^115 + 5*q^139 + 6*q^163 + 7*q^187 + ...

T. M. Apostol, An Introduction to Analytic Number Theory, Springer-Verlag, NY, 1976

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.
Noureddine Chair, The Euler-Riemann Gases, and Partition Identities, arXiv:1306.5415 [math-ph], 23-June-2013.
Donald Spector, Duality, partial supersymmetry and arithmetic number theory, arXiv:hep-th/9710002, 1997.
Donald Spector, Duality, partial supersymmetry and arithmetic number theory, J. Math. Phys. Vol. 39, 1998, p. 1919.

Cf. A047273, A056970, A097451, A098884.

a096981 = p $ tail a047273_list where
   p _  0         = 1
   p ks'@(k:ks) m = if k > m then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Feb 19 2013

series(product(1/(1-x^k+x^(2*k)-x^(3*k)+x^(4*k)-x^(5*k)), k=1..150), x=0,100);

CoefficientList[ Series[ Product[ 1/(1 - x^k + x^(2k) - x^(3k) + x^(4k) - x^(5k)), {k, 55}], {x, 0, 53}], x] (* Robert G. Wilson v, Aug 21 2004 *)
nmax = 100; CoefficientList[Series[x^3*QPochhammer[-1/x^2, x^3] * QPochhammer[-1/x, x^3]/((1 + x)*(1 + x^2) * QPochhammer[x^3, x^3]), {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 + A) * eta(x^6 + A)), n))} /* Michael Somos, Jun 08 2012 */

Expansion of q^(5/24) * eta(q^2) / (eta(q) * eta(q^6)) in powers of q. - Michael Somos, Jun 08 2012
Euler transform of period 6 sequence [1, 0, 1, 0, 1, 1, ...]. - Vladeta Jovovic, Aug 20 2004
G.f.: 1/product_{k>=1}(1-x^k+x^(2*k)-x^(3*k)+x^(4*k)-x^(5*k)) = Product_{k>=1}(1+x^(3*k-1))(1+x^(3*k-2))/(1-x^(3*k)).
a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*sqrt(6)*n). - Vaclav Kotesovec, Aug 31 2015

Better definition from Vladeta Jovovic, Aug 20 2004
More terms from Robert G. Wilson v, Aug 21 2004
Incorrect b-file replaced by Vaclav Kotesovec, Aug 31 2015

A263345 Expansion of Product_{k>=1} ((1 + x^k)/(1 + x^(3*k)))^k.

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 22, 40, 65, 107, 176, 282, 448, 705, 1101, 1701, 2611, 3977, 6021, 9048, 13527, 20102, 29720, 43712, 63997, 93259, 135317, 195539, 281440, 403559, 576568, 820888, 1164826, 1647583, 2323169, 3266041, 4578305, 6399990, 8922389, 12406535

Offset: 0

Vaclav Kotesovec, Oct 15 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
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

Cf. A003105, A026007, A262876, A262877, A262878, A262879, A263346.

Cf. A285289, A285290, A285291.

nmax=40; CoefficientList[Series[Product[((1 + x^k)/(1 + x^(3*k)))^k,{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ Zeta(3)^(1/6) * exp(2^(-1/3) * 3^(2/3) * Zeta(3)^(1/3) * n^(2/3)) / (2^(1/6) * 3^(2/3) * sqrt(Pi) * n^(2/3)).

A290307 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)/(1 + x^(k*j)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 2, 1, 2, 1, 0, 1, 1, 1, 2, 2, 2, 2, 1, 0, 1, 1, 1, 2, 2, 2, 3, 3, 2, 0, 1, 1, 1, 2, 2, 3, 3, 3, 3, 2, 0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 3, 2, 0, 1, 1, 1, 2, 2, 3, 4, 4, 4, 5, 4, 2, 0, 1, 1, 1, 2, 2, 3, 4, 4, 5, 6, 6, 5, 3, 0

Offset: 0

Ilya Gutkovskiy, Jul 26 2017

A(n,k) is the number of partitions of n into distinct parts where no part is a multiple of k.

			Square array begins:
  1,  1,  1,  1,  1,  1, ...
  0,  1,  1,  1,  1,  1, ...
  0,  0,  1,  1,  1,  1, ...
  0,  1,  1,  2,  2,  2, ...
  0,  1,  1,  1,  2,  2, ...
  0,  1,  2,  2,  2,  3, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to partitions

Columns k=1-10 give: A000007, A000700, A003105, A070048, A096938, A261770, A097793, A261771, A112193, A261772.

Cf. A286653.

Table[Function[k, SeriesCoefficient[Product[(1 + x^i)/(1 + x^(i k)), {i, Infinity}], {x, 0, n}]][j - n + 1], {j, 0, 13}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[QPochhammer[-1, x]/QPochhammer[-1, x^k], {x, 0, n}]][j - n + 1], {j, 0, 13}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} (1 + x^j)/(1 + x^(k*j)).

For asymptotics of column k see comment from Vaclav Kotesovec in A261772.

A058539 McKay-Thompson series of class 18d for the Monster group.

Original entry on oeis.org

1, 4, 10, 20, 35, 60, 100, 164, 261, 400, 600, 884, 1291, 1864, 2656, 3740, 5205, 7184, 9842, 13388, 18082, 24244, 32300, 42784, 56378, 73928, 96466, 125284, 161981, 208568, 267524, 341880, 435343, 552424, 698666, 880848, 1107229, 1387804, 1734624, 2162248

Offset: 0

N. J. A. Sloane, Nov 27 2000

nonn

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

			1 + 4*x + 10*x^2 + 20*x^3 + 35*x^4 + 60*x^5 + 100*x^6 + 164*x^7 + ...
T18d = 1/q + 4*q^2 + 10*q^5 + 20*q^8 + 35*q^11 + 60*q^14 + 100*q^17 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for McKay-Thompson series for Monster simple group

Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.

Cf. A003105, A103262.

nmax = 50; CoefficientList[Series[Product[((1+x^(3*k-1))*(1+x^(3*k-2)))^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 10 2015 *)
QP = QPochhammer; s = (QP[q^2]*(QP[q^3]/(QP[q]*QP[q^6])))^4 + 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^2 + A) * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)))^4, n))} /* Michael Somos, Mar 04 2012 */

Expansion of (chi(-x^3) / chi(-x))^4 in powers of x where chi() is a Ramanujan theta function.
Expansion of q^(1/3) * c(q) * b(q^2) / (b(q) * c(q^2)) in powers of q where b(), c() are cubic AGM theta functions.
Expansion of q^(1/3) * (eta(q^2) * eta(q^3) / (eta(q) * eta(q^6)))^4 in powers of q.
Given g.f. A(x), then B(x) = A(x^3) / x satisfies 0 = f(B(x), B(x^2)) where f(u, v) = 8 * (u * v)^2 - (1 + u * v) * (u^2 - v) * (v^2 - u).
Given g.f. A(x), then B(x) = A(x^3) / x satisfies 0 = f(B(x), B(x^2)) where f(u, v) = 9 * (u * v)^2 - (u - v^2 + u^2*v) * (v - u^2 + u*v^2).
Given g.f. A(x), then B(x) = A(x^3) / x satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = 8 * u * v * w - (u^2 - v) * (w^2 - v).
Given g.f. A(x), then B(x) = A(x^3) / x satisfies 0 = f(B(x), B(x^5)) where f(u, v) = u*v * (1 + 25 * u*v + u^2*v^2)^2 - (u^3 + v^3 + 10 * u*v * (1 + u*v))^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (54 t)) = f(t) where q = exp(2 Pi i t).
Convolution square of A103262. Convolution fourth power of A003105.
a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(3/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Sep 10 2015

A103262 McKay-Thompson series of class 36g for the Monster group.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 11, 16, 21, 26, 34, 44, 58, 74, 93, 116, 143, 178, 221, 272, 332, 402, 487, 588, 710, 854, 1021, 1216, 1444, 1714, 2031, 2400, 2826, 3318, 3888, 4552, 5322, 6208, 7224, 8388, 9726, 11264, 13028, 15044, 17339, 19952, 22930, 26324, 30186

Offset: 0

Noureddine Chair, Feb 21 2005

nonn

Number of partitions of n into distinct parts prime to 3, with 2 types of each part.

This is also the number of partitions of n into parts with 2 types congruent to 1 or 5 mod(6).

			E.g., a(5)=8 because we have 5,5*,41,41*,4*1,4*1*,22*1,22*1* with all parts prime to 3. The parts congruent to 1,5 mod(6) are 5, 5*, 11111, 11111*, 1111*1*, 111*1*1*, 11*1*1*1*, 1*1*1*1*1*.
T36g = 1/q + 2*q^5 + 3*q^11 + 4*q^17 + 5*q^23 + 8*q^29 + 11*q^35 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
Index entries for McKay-Thompson series for Monster simple group

Cf. A003105.

series(product((1+x^k)^2/(1+x^(3*k))^2,k=1..100),x=0,100);

CoefficientList[ Series[ Product[(1 + x^k)^2/(1 + x^(3k))^2, {k, 60}], {x, 0, 50}], x] (* Robert G. Wilson v, Feb 22 2005 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(1/6)(eta[q^2]eta[q^3]/(eta[q]eta[q^6]))^2, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 06 2018 *)

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)*eta(x^3+A)/eta(x+A)/eta(x^6+A))^2, n))} /* Michael Somos, Sep 10 2005 */

G.f.: product_{k>0}((1+x^k)/(1+x^(3k)))^2= 1/product_{k>0}((1-x^(6k-1))*(1-x^(6k-5)))^2.
Expansion of q^(1/6)(eta(q^2)eta(q^3)/(eta(q)eta(q^6)))^2 in powers of q.
Euler transform of period 6 sequence [2, 0, 0, 0, 2, 0, ...]. - Michael Somos, Sep 10 2005
a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Sep 01 2015

More terms from Robert G. Wilson v, Feb 22 2005

cp's OEIS Frontend

A261771 Expansion of Product_{k>=1} (1 + x^k) / (1 + x^(8*k)).

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A261772 Expansion of Product_{k>=1} (1 + x^k) / (1 + x^(10*k)).

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A003113 Coefficients in expansion of permanent of infinite tridiagonal matrix shown below.

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A132463 Number of partitions of n into distinct parts congruent to 0 or 1 modulo 3.

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A132462 Number of partitions of n into distinct parts congruent to 0 or 2 modulo 3.

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A096981 Number of partitions of n into parts congruent to {0, 1, 3, 5} mod 6.

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A263345 Expansion of Product_{k>=1} ((1 + x^k)/(1 + x^(3*k)))^k.

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