A351342 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 * A(x/(1 - 2*x)) / (1 - 2*x).

1, 1, 1, 1, 3, 9, 27, 83, 271, 971, 3865, 16879, 78985, 388385, 1987201, 10561385, 58443891, 337724057, 2040085491, 12862712499, 84357800063, 573182197539, 4021203303593, 29062345301487, 216129411635057, 1653180368063361, 13003920016983361, 105158133803473329

Offset: 0

Author

Ilya Gutkovskiy, Feb 08 2022

Keywords

• nonn

Comments

Shifts 3 places left under 2nd-order binomial transform.

Crossrefs

Cf. A000996, A000998, A004211, A007472, A210540, A351343, A351344, A351345.

Programs

Mathematica

nmax = 27; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 A[x/(1 - 2 x)]/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = If[n < 3, 1, Sum[Binomial[n - 3, k] 2^k a[n - k - 3], {k, 0, n - 3}]]; Table[a[n], {n, 0, 27}]

Formula

a(0) = a(1) = a(2) = 1; a(n) = Sum_{k=0..n-3} binomial(n-3,k) * 2^k * a(n-k-3).