A307709 G.f. A(x) satisfies: A(x) = x*exp(Sum_{n>=1} Sum_{k>=1} (-1)^(k+1)*n*a(n)*x^(n*k)/k).

0, 1, 1, 2, 8, 39, 240, 1726, 14189, 130790, 1334873, 14935261, 181757070, 2390363103, 33787482806, 510879775905, 8229231626244, 140698662551489, 2545031258526411, 48561748888174242, 974858528384809324, 20539450727358618457, 453191882910044604304, 10450775091071489910630

Offset

0,4

Links

Table of n, a(n) for n=0..23.

Formula

G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} (1 + x^n)^(n*a(n)).

Recurrence: a(n+1) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) d^2*a(d) ) * a(n-k+1).

Example

G.f.: A(x) = x + x^2 + 2*x^3 + 8*x^4 + 39*x^5 + 240*x^6 + 1726*x^7 + 14189*x^8 + 130790*x^9 + 1334873*x^10 + ...

Mathematica

a[n_] := a[n] = SeriesCoefficient[x Exp[Sum[Sum[(-1)^(k + 1) j a[j] x^(j k)/k, {k, 1, n - 1}], {j, 1, n - 1}]], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 0, 23}]
a[n_] := a[n] = SeriesCoefficient[x Product[(1 + x^k)^(k a[k]), {k, 1, n - 1}], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 0, 23}]

Crossrefs

Cf. A004111, A307708.

Sequence in context: A347665 A068107 A308207 * A217945 A091073 A266468

Adjacent sequences: A307706 A307707 A307708 * A307710 A307711 A307712

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 23 2019

Status

approved