A261630 -id:A261630 - cp's OEIS frontend

A261634 Expansion of Product_{k>=0} (1+x^(4*k+1))^3.

1, 3, 3, 1, 0, 3, 9, 9, 3, 3, 12, 18, 12, 6, 18, 37, 33, 15, 22, 54, 66, 42, 36, 81, 114, 84, 57, 112, 189, 171, 109, 156, 279, 294, 201, 222, 405, 486, 360, 328, 564, 747, 617, 504, 783, 1123, 1017, 783, 1065, 1602, 1605

Offset: 0

Vaclav Kotesovec, Aug 27 2015

nonn

Cf. A261612, A261615, A261633, A169975, A261630.

nmax=50; CoefficientList[Series[Product[(1+x^(4*k+1))^3, {k, 0, nmax}], {x, 0, nmax}], x]

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

A261638 Expansion of Product_{k>=0} (1+x^(4*k+1))^4.

Original entry on oeis.org

1, 4, 6, 4, 1, 4, 16, 24, 16, 8, 22, 48, 52, 32, 38, 92, 128, 96, 70, 140, 245, 244, 172, 228, 417, 488, 374, 380, 680, 924, 798, 676, 1044, 1560, 1542, 1256, 1625, 2524, 2778, 2304, 2537, 3892, 4716, 4156, 4076, 5908, 7650, 7196, 6592, 8796, 11938

Offset: 0

Vaclav Kotesovec, Aug 27 2015

nonn

In general, if j > 0, a > 0, b > 0, GCD(a,b) = 1 and g.f. = Product_{k>=0} (1 + x^(a*k+b))^j, then a(n) ~ 2^((j-3)/2 - j*b/a) * j^(1/4) * exp(Pi*sqrt(j*n/(3*a))) / ((3*a)^(1/4) * n^(3/4)).

Cf. A261612, A261615, A261637, A169975, A261630, A261634.

nmax=50; CoefficientList[Series[Product[(1+x^(4*k+1))^4, {k, 0, nmax}], {x, 0, nmax}], x]

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

cp's OEIS Frontend

A261634 Expansion of Product_{k>=0} (1+x^(4*k+1))^3.

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