A343200 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+3,3).

1, 4, 16, 64, 221, 736, 2338, 7132, 21093, 60652, 170172, 467140, 1257571, 3325824, 8654576, 22189340, 56116043, 140122760, 345769094, 843827436, 2038017983, 4874329024, 11550814704, 27134195608, 63215468883, 146120097736, 335227455982, 763592477104, 1727482413548

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..7650

Formula

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) * A033488(d) ) * a(n-k).

a(n) ~ (3*zeta(5))^(1/10) / (2^(7/10) * 5^(2/5) * sqrt(Pi) * n^(3/5)) * exp(-469*log(2)/720 - 2401*Pi^16 / (656100000000*zeta(5)^3) + 539*Pi^8*zeta(3) / (8100000*zeta(5)^2) - 7*Pi^6 / (27000*zeta(5)) - 121*zeta(3)^2 / (600*zeta(5)) + (343*Pi^12 / (303750000 * 2^(3/5) * 15^(1/5) * zeta(5)^(11/5)) - 77*Pi^4*zeta(3) / (4500 * 2^(3/5) * 15^(1/5) * zeta(5)^(6/5)) + Pi^2 / (6*2^(3/5) * (15*zeta(5))^(1/5))) * n^(1/5) + (-49*Pi^8 / (270000 * 2^(1/5) * 15^(2/5) * zeta(5)^(7/5)) + 11*zeta(3) / (4*2^(1/5) * (15*zeta(5))^(2/5))) * n^(2/5) + (7*Pi^4 / (90*2^(4/5) * (15*zeta(5))^(3/5))) * n^(3/5) + (5*(15*zeta(5))^(1/5) / (4*2^(2/5))) * n^(4/5)). - Vaclav Kotesovec, May 12 2021

Mathematica

nmax = 28; CoefficientList[Series[Product[(1 + x^k)^Binomial[k + 3, 3], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d Binomial[d + 3, 3], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 28}]

Crossrefs

Cf. A000292, A033488, A219555, A255050, A258343, A338645, A344097, A344098.

Sequence in context: A249567 A348906 A177398 * A227312 A267729 A189154

Adjacent sequences: A343197 A343198 A343199 * A343201 A343202 A343203

Keyword

nonn

Author

Ilya Gutkovskiy, May 09 2021

Status

approved