A346051 G.f. A(x) satisfies: A(x) = 1 + x^2 + x^3 * A(x/(1 - x)) / (1 - x).

1, 0, 1, 1, 1, 2, 5, 12, 28, 68, 181, 531, 1671, 5491, 18627, 65299, 237880, 903907, 3580619, 14729777, 62639952, 274442521, 1236730244, 5729809348, 27292248240, 133614280479, 671803041553, 3464970976743, 18309428363425, 99010800275743, 547462187824465, 3093329527120022

Offset: 0

Author

Ilya Gutkovskiy, Jul 02 2021

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..695

Crossrefs

Cf. A000994, A000995, A000996, A000997, A000998.

Cf. A007476, A210540, A346050, A346052.

Programs

Magma

function a(n)
  if n lt 3 then return (1+(-1)^n)/2;
  else return (&+[Binomial(n-3,j)*a(j): j in [0..n-3]]);
  end if; return a;
end function;
[a(n): n in [0..35]]; // G. C. Greubel, Nov 30 2022

Mathematica

nmax = 31; A[] = 0; Do[A[x] = 1 + x^2 + x^3 A[x/(1 - x)]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 0; a[2] = 1; a[n_] := a[n] = Sum[Binomial[n - 3, k] a[k], {k, 0, n - 3}]; Table[a[n], {n, 0, 31}]

SageMath

@CachedFunction
def a(n): # a = A346051
    if (n<3): return (1, 0, 1)[n]
    else: return sum(binomial(n-3, k)*a(k) for k in range(n-2))
[a(n) for n in range(51)] # G. C. Greubel, Nov 30 2022

Formula

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