A294839 Expansion of Product_{k>=1} (1 + x^(2*k-1))^(k*(3*k-1)/2)*(1 + x^(2*k))^(k*(3*k+1)/2).

1, 1, 2, 7, 13, 30, 61, 125, 250, 494, 960, 1835, 3487, 6520, 12105, 22239, 40515, 73207, 131315, 233831, 413625, 727100, 1270405, 2207243, 3814155, 6557164, 11217391, 19099932, 32375026, 54640509, 91836697, 153739008, 256379360, 425964293, 705197513, 1163452547, 1913096832, 3135609791

Offset

0,3

Comments

Weigh transform of the generalized pentagonal numbers (A001318).

Links

Table of n, a(n) for n=0..37.

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

N. J. A. Sloane, Transforms

Eric Weisstein's World of Mathematics, Pentagonal Number

Formula

G.f.: Product_{k>=1} (1 + x^k)^A001318(k).

a(n) ~ exp(Pi*sqrt(2) * 7^(1/4) * n^(3/4) / (3*5^(1/4)) + 9*Zeta(3) * sqrt(5*n/7) / (4*Pi^2) + (7*Pi^6 - 2430*Zeta(3)^2) * (5/7)^(1/4) * n^(1/4) / (336 * sqrt(2) * Pi^5) + 15*Zeta(3)*(3240*Zeta(3)^2 - 7*Pi^6) / (3136*Pi^8)) * 7^(1/8) / (2^(9/4) * 5^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 10 2017

Mathematica

nmax = 37; CoefficientList[Series[Product[(1 + x^(2 k - 1))^(k (3 k - 1)/2) (1 + x^(2 k))^(k (3 k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d Ceiling[d/2] Ceiling[(3 d + 1)/2]/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 37}]

Crossrefs

Cf. A001318, A028377, A294102, A294840, A294841.

Sequence in context: A215206 A298215 A127396 * A231478 A320680 A079119

Adjacent sequences: A294836 A294837 A294838 * A294840 A294841 A294842

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 09 2017

Status

approved