A147599 Expansion of Product_{i>=1} (1+x^(4*i-1)).

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

Offset: 0

Author

N. J. A. Sloane, Aug 29 2010

Keywords

• nonn

Comments

Number of partitions into distinct parts 4*k+3.

Convolution of A147599 and A169975 is A000700. - Vaclav Kotesovec, Jan 18 2017

Links

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

Crossrefs

Cf. A000700, A169975, A170956-A170965.

Cf. A262928, A281243, A281244, A281245.

Programs

Mathematica

nmax = 200; CoefficientList[Series[Product[(1 + x^(4*k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 18 2017 *)
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 4] == 3, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 18 2017 *)

Formula

G.f. sum(n>=0, x^(2*n^2+n) / prod(k=1,n, 1-x^(4*k))) - Joerg Arndt, Mar 10 2011.

a(n) ~ exp(sqrt(n/3)*Pi/2) / (4*6^(1/4)*n^(3/4)) * (1 - (3*sqrt(3)/(4*Pi) + Pi/(192*sqrt(3))) / sqrt(n)). - Vaclav Kotesovec, Jan 18 2017