A023423 Generalized Catalan Numbers.

1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 257, 517, 1042, 2104, 4256, 8624, 17504, 35585, 72455, 147746, 301706, 616948, 1263240, 2589840, 5316033, 10924681, 22475831, 46290195

Offset

0,8

Links

Seiichi Manyama, Table of n, a(n) for n = 0..3028

Formula

G.f. A(x) satisfies: A(x) = (1 + x^2 * A(x)^2) / (1 - x + x^2 + x^3 + x^4 + x^5 + x^6). - Ilya Gutkovskiy, Jul 20 2021

Maple

A023423 := proc(n)
    option remember;
    if n <= 6 then
        1;
    else
        procname(n-1)+add(procname(k)*procname(n-2-k), k=5..n-2) ;
    end if;
end proc: # R. J. Mathar, Oct 10 2014

Mathematica

a[0]=1; a[n_]:= a[n]=a[n-1] + Sum[a[k]*a[n-2-k], {k, 5, n-2}]; Table[a[n], {n, 0, 30}] (* modified by G. C. Greubel, Jan 01 2018 *)

Prog

(PARI) {a(n) = if(n==0, 1, a(n-1) + sum(k=5, n-2, a(k)*a(n-k-2)))};
for(n=0, 30, print1(a(n), ", ")) \\ G. C. Greubel, Jan 01 2018

Crossrefs

Cf. A000108, A001006, A004148, A004149, A023421, A023422.

Sixth row of A064645.

Sequence in context: A275063 A186026 A367653 * A210544 A180210 A274861

Adjacent sequences: A023420 A023421 A023422 * A023424 A023425 A023426

Keyword

nonn,easy

Author

Olivier Gérard

Status

approved