A346073 a(n) = 1 + Sum_{k=0..n-4} a(k) * a(n-k-4).

1, 1, 1, 1, 2, 3, 4, 5, 8, 13, 20, 29, 45, 73, 118, 185, 293, 475, 778, 1263, 2047, 3345, 5512, 9085, 14957, 24683, 40918, 67987, 113016, 188053, 313608, 524041, 876657, 1467797, 2460644, 4130893, 6942726, 11678687, 19663068, 33139295, 55904339, 94384167, 159470488

Offset: 0

Author

Ilya Gutkovskiy, Jul 04 2021

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000108, A007317, A090344, A216604, A307971, A346074.

Programs

Mathematica

a[n_] := a[n] = 1 + Sum[a[k] a[n - k - 4], {k, 0, n - 4}]; Table[a[n], {n, 0, 42}]
nmax = 42; A[] = 0; Do[A[x] = 1/(1 - x) + x^4 A[x]^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

PARI

a(n) = sum(k=0, n\4, binomial(n-3*k, k)*binomial(2*k, k)/(k+1)); \\ Seiichi Manyama, Jan 22 2023

SageMath

@CachedFunction
def a(n): # a = A346073
    if (n<4): return 1
    else: return 1 + sum(a(k)*a(n-k-4) for k in range(n-3))
[a(n) for n in range(51)] # G. C. Greubel, Nov 26 2022

Formula

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

a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k,k) * Catalan(k). - Seiichi Manyama, Jan 22 2023