A345243 G.f. A(x) satisfies: A(x) = x + x^2 * exp(2 * Sum_{k>=1} (-1)^(k+1) * A(x^k) / k).

1, 1, 2, 3, 8, 17, 42, 107, 272, 719, 1914, 5163, 14088, 38733, 107370, 299511, 840372, 2370020, 6714316, 19100096, 54534696, 156230943, 448942998, 1293692305, 3737568960, 10823759093, 31413810702, 91358248179, 266193726712, 776989772307, 2271695757714, 6652074198889

Offset

1,3

Links

Table of n, a(n) for n=1..32.

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} (-1)^(k/d+1) * d * a(d) ) * a(n-k+2).

Mathematica

nmax = 32; A[_] = 0; Do[A[x_] = x + x^2 Exp[2 Sum[(-1)^(k + 1) 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[(-1)^(k/d + 1) d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 32}]

Crossrefs

Cf. A005753, A007560, A345244, A345245.

Sequence in context: A099965 A319566 A294450 * A292852 A148014 A148015

Adjacent sequences: A345240 A345241 A345242 * A345244 A345245 A345246

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 11 2021

Status

approved