A294844 Expansion of Product_{k>=1} (1 + x^k)^(k*(k+1)*(5*k-2)/6).
1, 1, 8, 34, 114, 411, 1380, 4573, 14650, 45995, 141296, 426364, 1265443, 3698011, 10657134, 30312395, 85183177, 236681860, 650686538, 1771098691, 4775571943, 12762628737, 33821018537, 88909273699, 231945942992, 600700301298, 1544897610261, 3946762859175, 10018454809275, 25274880698255
Offset: 0
Keywords
Links
- 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 Pyramidal Number
Programs
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Mathematica
nmax = 29; CoefficientList[Series[Product[(1 + x^k)^(k (k + 1) (5 k - 2)/6), {k, 1, nmax}], {x, 0, nmax}], x] a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^2 (d + 1) (5 d - 2)/6, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 29}]
Formula
G.f.: Product_{k>=1} (1 + x^k)^A002413(k).
a(n) ~ (3*Zeta(5))^(1/10) / (2^(479/720) * 5^(3/10) * sqrt(Pi) * n^(3/5)) * exp(-2401 * Pi^16 / (1312200000000000 * Zeta(5)^3) - 49 * Pi^8 * Zeta(3) / (405000000 * Zeta(5)^2) - Zeta(3)^2 / (750*Zeta(5)) + (343*Pi^12 / (60750000000 * 2^(3/5) * 3^(1/5) * 5^(2/5) * Zeta(5)^(11/5)) + 7*Pi^4 * Zeta(3) / (22500 * 2^(3/5) * 3^(1/5) * 5^(2/5) * Zeta(5)^(6/5))) * n^(1/5) - (49*Pi^8 / (5400000 * 2^(1/5) * 3^(2/5) * 5^(4/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(6/5) * 5^(4/5) * (3*Zeta(5))^(2/5))) * n^(2/5) + (7*Pi^4 / (900 * 2^(4/5) * 5^(1/5) * (3*Zeta(5))^(3/5))) * n^(3/5) + (5^(7/5) * (3*Zeta(5))^(1/5) / 2^(12/5)) * n^(4/5)). - Vaclav Kotesovec, Nov 10 2017
Comments