A345200 G.f. A(x) satisfies: A(x) = x + x^2 * exp(2 * Sum_{k>=1} A(x^k) / k).

1, 1, 2, 5, 12, 32, 84, 234, 652, 1872, 5416, 15922, 47188, 141283, 425910, 1293105, 3948080, 12118619, 37367694, 115708111, 359623780, 1121543440, 3508533500, 11006973980, 34620982004, 109157354769, 344928572562, 1092190467567, 3464955417200, 11012117992012

Offset

1,3

Links

Table of n, a(n) for n=1..30.

Formula

G.f.: x + x^2 / Product_{n>=1} (1 - x^n)^(2*a(n)).

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

a(n) ~ c * d^n / n^(3/2), where d = 3.3437762102302517833309792925121217026126033230718263962128740290952197... and c = 0.3397354606156870289877990463189432389789387070060129709272911771... - Vaclav Kotesovec, Jun 19 2021

Mathematica

nmax = 30; A[_] = 0; Do[A[x_] = x + x^2 Exp[2 Sum[A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = a[2] = 1; a[n_] := a[n] = (2/(n - 2)) Sum[Sum[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 30}]

Crossrefs

Cf. A000151, A007562, A345241, A345242.

Sequence in context: A277863 A039809 A335456 * A173611 A292211 A293348

Adjacent sequences: A345197 A345198 A345199 * A345201 A345202 A345203

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 11 2021

Status

approved