A294840 Expansion of Product_{k>=1} (1 + x^(2*k-1))^(k*(5*k-3)/2)*(1 + x^(2*k))^(k*(5*k+3)/2).

1, 1, 4, 11, 26, 65, 150, 343, 760, 1670, 3574, 7561, 15752, 32396, 65850, 132386, 263447, 519316, 1014744, 1966234, 3780464, 7215020, 13674227, 25744768, 48166429, 89576421, 165638008, 304615115, 557275053, 1014398476, 1837617957, 3313527482, 5948262037, 10632231253, 18926026208

Offset

0,3

Comments

Weigh transform of the generalized heptagonal numbers (A085787).

Links

Table of n, a(n) for n=0..34.

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, Heptagonal Number

Formula

G.f.: Product_{k>=1} (1 + x^k)^A085787(k).

a(n) ~ 7^(1/8) * exp(Pi*sqrt(2) * 7^(1/4) * n^(3/4) / 3^(5/4) + 15*Zeta(3) * sqrt(3*n/7) / (4*Pi^2) - (7*Pi^6 + 4050*Zeta(3)^2)*n^(1/4) / (112*sqrt(2) * 3^(3/4) * 7^(1/4) * Pi^5) + 15*Zeta(3) * (7*Pi^6 + 5400*Zeta(3)^2) / (3136*Pi^8)) / (2^(7/3) * 3^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 10 2017

Mathematica

nmax = 34; CoefficientList[Series[Product[(1 + x^(2 k - 1))^(k (5 k - 3)/2) (1 + x^(2 k))^(k (5 k + 3)/2), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (5 d (d + 1)/8 + (-1)^d (2 d + 1)/16 - 1/16), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 34}]

Crossrefs

Cf. A028377, A085787, A294837, A294839, A294841.

Sequence in context: A183276 A340567 A268775 * A014630 A192965 A305119

Adjacent sequences: A294837 A294838 A294839 * A294841 A294842 A294843

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 09 2017

Status

approved