A363467 G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} (-1)^(k+1) * A(x^k)^3 / (k*x^(2*k)) ).

1, 1, 1, 3, 9, 25, 88, 292, 1031, 3685, 13433, 49608, 185465, 699963, 2664650, 10217130, 39428179, 153009240, 596761737, 2337875430, 9195732624, 36301739221, 143780858517, 571191310205, 2275409450019, 9087376470138, 36377539265376, 145937953205705, 586645566919856

Offset: 1

Author

Ilya Gutkovskiy, Jun 03 2023

Keywords

• nonn

Crossrefs

Cf. A007560, A052755, A363388, A363465, A363468.

Programs

Mathematica

nmax = 29; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[(-1)^(k + 1) A[x^k]^3/(k x^(2 k)), {k, 1, nmax}]] + O[x]^(nmax + 1)//Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = a[2] = 1; f[n_] := f[n] = Sum[a[k] a[n - k], {k, 1, n - 1}]; g[n_] := g[n] = Sum[a[k] f[n - k], {k, 1, n - 1}]; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[(-1)^(k/d + 1) d g[d + 2], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 29}]