A305204 Expansion of Product_{k>=1} 1/(1 - (k*(k + 1)/2)*x^k).

1, 1, 4, 10, 29, 62, 176, 363, 931, 2029, 4751, 10062, 23749, 48959, 109342, 230981, 500344, 1031667, 2223218, 4531585, 9570395, 19523510, 40411313, 81628389, 168484616, 336850254, 685112670, 1369559157, 2757908932, 5464925114, 10958578421, 21574592680

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..3841

Formula

G.f.: Product_{k>=1} 1/(1 - A000217(k)*x^k).

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (j*(j + 1))^k*x^(j*k)/(k*2^k)).

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1,
      1, b(n, i-1)+(1+i)*i/2*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..33);  # Alois P. Heinz, Aug 16 2019

Mathematica

nmax = 31; CoefficientList[Series[Product[1/(1 - (k (k + 1)/2) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 31; CoefficientList[Series[Exp[Sum[Sum[(j (j + 1))^k x^(j k)/(k 2^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^(k/d + 1) ((d + 1)/2)^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 31}]

Crossrefs

Cf. A000217, A000294, A006906, A077335, A265836.

Sequence in context: A111236 A164361 A006907 * A327590 A321344 A329156

Adjacent sequences: A305201 A305202 A305203 * A305205 A305206 A305207

Keyword

nonn

Author

Ilya Gutkovskiy, May 27 2018

Status

approved