A346051 -id:A346051 - cp's OEIS frontend

A346050 G.f. A(x) satisfies: A(x) = x + x^2 + x^3 * A(x/(1 - x)) / (1 - x).

0, 1, 1, 0, 1, 3, 6, 11, 23, 60, 179, 553, 1716, 5415, 17801, 61956, 228391, 882309, 3530322, 14531621, 61454091, 267479778, 1200680113, 5561767211, 26553471186, 130366882251, 656668581417, 3387887246292, 17886582294921, 96603394562849, 533645344137390, 3014295344076655

Offset: 0

Ilya Gutkovskiy, Jul 02 2021

nonn

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

Cf. A000994, A000995, A000996, A000997, A000998, A007476, A210540, A346051, A346052.

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

@CachedFunction
def a(n): # a = A346050
    if (n<3): return (0,1,1)[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 28 2022

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

A346052 G.f. A(x) satisfies: A(x) = 1 + x + x^3 * A(x/(1 - x)) / (1 - x).

Original entry on oeis.org

1, 1, 0, 1, 2, 3, 5, 11, 29, 80, 222, 630, 1881, 6004, 20420, 72979, 270659, 1035590, 4087205, 16675630, 70440641, 307933393, 1390117953, 6462787357, 30871458702, 151298796000, 760250325004, 3915477534861, 20662363081756, 111662169790416, 617482470676567, 3490973387652861

Offset: 0

Ilya Gutkovskiy, Jul 02 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 0..650

Cf. A000994, A000995, A000996, A000997, A000998, A007476, A210540, A346050, A346051.

function a(n) // a = A346052
  if n lt 3 then return Floor((3-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

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

@CachedFunction
def a(n): # a = A346052
    if (n<3): return (1, 1, 0)[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

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

cp's OEIS Frontend

A346050 G.f. A(x) satisfies: A(x) = x + x^2 + x^3 * A(x/(1 - x)) / (1 - x).

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