%I #7 Nov 10 2017 10:11:14
%S 1,1,7,29,93,320,1026,3256,9995,30102,88722,257042,732876,2058370,
%T 5703858,15606076,42203027,112882223,298849221,783574536,2035876825,
%U 5244191462,13398463986,33967008194,85476285603,213583335753,530099612487,1307195997381,3203555001240,7804386224233
%N Expansion of Product_{k>=1} (1 + x^k)^(k*(k+1)*(4*k-1)/6).
%C Weigh transform of the hexagonal pyramidal numbers (A002412).
%H M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HexagonalPyramidalNumber.html">Hexagonal Pyramidal Number</a>
%F G.f.: Product_{k>=1} (1 + x^k)^A002412(k).
%F a(n) ~ exp(-2401 * Pi^16 / (671846400000000 * Zeta(5)^3) - 49*Pi^8 * Zeta(3) / (518400000 * Zeta(5)^2) - Zeta(3)^2 / (2400*Zeta(5)) + (343 * Pi^12 / (77760000000 * 15^(1/5) * Zeta(5)^(11/5)) + 7*Pi^4*Zeta(3) / (72000 * 15^(1/5) * Zeta(5)^(6/5))) * n^(1/5) - (49*Pi^8 / (8640000 * 15^(2/5) * Zeta(5)^(7/5)) + Zeta(3) / (8 * (15*Zeta(5))^(2/5))) * n^(2/5) + (7*Pi^4 / (720 * (15*Zeta(5))^(3/5))) * n^(3/5) + (5*(15*Zeta(5))^(1/5)/4) * n^(4/5)) * (3*Zeta(5))^(1/10) / (2^(173/360) * 5^(2/5) * sqrt(Pi) * n^(3/5)). - _Vaclav Kotesovec_, Nov 10 2017
%t nmax = 29; CoefficientList[Series[Product[(1 + x^k)^(k (k + 1)(4 k - 1)/6), {k, 1, nmax}], {x, 0, nmax}], x]
%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^2 (d + 1)(4 d - 1)/6, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 29}]
%Y Cf. A002412, A028377, A258343, A281156, A294836, A294842, A294844, A294845.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Nov 09 2017