A349014 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x) / (1 - x) + x^2 * A(x)^2.

1, 1, 2, 4, 9, 20, 47, 111, 270, 663, 1656, 4174, 10636, 27308, 70651, 183902, 481436, 1266515, 3346793, 8879116, 23642034, 63156917, 169222939, 454660940, 1224650739, 3306338583, 8945780742, 24252558183, 65872671839, 179228552638, 488443704486

Offset: 0

Author

Ilya Gutkovskiy, Nov 05 2021

Keywords

• nonn

Crossrefs

Cf. A007477, A215973, A349015, A349016.

Programs

Mathematica

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

Formula

a(0) = a(1) = 1; a(n) = Sum_{k=0..n-2} a(k) * (1 + a(n-k-2)).

a(n) ~ sqrt(1/r + (2-r)*s/(1-r)^2 + 2*s^2) / (2*sqrt(Pi)*n^(3/2)*r^n), where r = 0.3495518575342322867499973927570340375314361958565... and s = 3.323404276086477625771682790702806844309937221726... are real roots of the system of equations 1 + r + r^2*s*(1/(1-r) + s) = s, r^2*(1/(1-r) + 2*s) = 1. - Vaclav Kotesovec, Nov 06 2021