A351128 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 4*x)) / (1 - 4*x).

0, 1, 0, 1, 8, 49, 280, 1649, 10800, 81505, 696400, 6472033, 63562872, 652984977, 7026210728, 79547049681, 949709767904, 11936248012993, 157219119485216, 2159448120457409, 30811324011852136, 455635009201780977, 6975424580445456056, 110478282815356437809

Offset: 0

Author

Ilya Gutkovskiy, Feb 03 2022

Keywords

• nonn

Comments

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

Crossrefs

Cf. A000995, A004213, A351028, A351050, A351053, A351132, A351150, A351161.

Programs

Mathematica

nmax = 23; A[] = 0; Do[A[x] = x + x^2 A[x/(1 - 4 x)]/(1 - 4 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 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, 23}]

Formula

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