A363386 G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} (-1)^(k+1) * A(x^k)^2 / k ).

1, 1, 0, 1, 2, 1, 4, 8, 10, 22, 50, 77, 160, 343, 622, 1250, 2648, 5127, 10364, 21685, 43594, 88907, 185458, 380113, 782902, 1633841, 3387444, 7033401, 14716304, 30734066, 64228198, 134824862, 283040684, 594516622, 1252151812, 2639220817, 5566237724, 11760037378

Offset

1,5

Links

Table of n, a(n) for n=1..38.

Mathematica

nmax = 38; A[_] = 0; Do[A[x_] = x + x^2 Exp[Sum[(-1)^(k + 1) A[x^k]^2/k, {k, 1, nmax}]] + O[x]^(nmax + 1)//Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = a[2] = 1; g[n_] := g[n] = Sum[a[k] a[n - k], {k, 1, n - 1}]; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[(-1)^(k/d + 1) d g[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 38}]

Prog

(PARI) seq(n)=my(p=x+x^2+O(x^3)); for(n=1, n\2, my(m=serprec(p, x)-1); p = x + x^2*exp(-sum(k=1, m\2, (-1)^k*subst(p + O(x^(m\k+1)), x, x^k)^2/k))); Vec(p + O(x*x^n)) \\ Andrew Howroyd, May 30 2023

Crossrefs

Cf. A005754, A007560, A363385.

Sequence in context: A353661 A156817 A008301 * A294104 A294061 A113820

Adjacent sequences: A363383 A363384 A363385 * A363387 A363388 A363389

Keyword

nonn

Author

Ilya Gutkovskiy, May 30 2023

Status

approved