A351050 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 4*x)) / (1 - 4*x).

1, 1, 1, 5, 25, 129, 713, 4373, 30289, 235041, 1998001, 18226117, 176364969, 1803064033, 19463340729, 221691818005, 2658751147297, 33458500940993, 440140082161121, 6032572875160069, 85936355674437561, 1270176766188103105, 19453176663852208937

Offset: 0

Author

Ilya Gutkovskiy, Jan 30 2022

Keywords

• nonn

Comments

Shifts 2 places left under 4th-order binomial transform.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..516

Crossrefs

Cf. A004213, A007472, A007476, A351049.

Programs

Mathematica

nmax = 22; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 - 4 x)]/(1 - 4 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] 4^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 22}]

Formula

a(0) = a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 4^k * a(n-k-2).