A360047 a(n) = Sum_{k=0..floor(n/5)} binomial(n+4,5*k+4) * Catalan(k).

1, 5, 15, 35, 70, 127, 220, 385, 715, 1430, 3005, 6400, 13500, 28050, 57800, 119515, 250425, 533525, 1151975, 2504700, 5453176, 11856275, 25748450, 55962300, 121981725, 266968938, 586630515, 1292992795, 2855288480, 6311930460, 13963767356, 30919563310

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

a(n) = binomial(n+4,4) + Sum_{k=0..n-5} a(k) * a(n-k-5).

G.f. A(x) satisfies: A(x) = 1/(1-x)^5 + x^5 * A(x)^2.

G.f.: 2 / ( (1-x)^2 * ((1-x)^3 + sqrt((1-x)^6 - 4*x^5*(1-x))) ).

D-finite with recurrence (n+5)*a(n) +6*(-n-4)*a(n-1) +15*(n+3)*a(n-2) +20*(-n-2)*a(n-3) +15*(n+1)*a(n-4) +10*(-n+1)*a(n-5) +5*(n-1)*a(n-6)=0. - R. J. Mathar, Jan 25 2023

Prog

(PARI) a(n) = sum(k=0, n\5, binomial(n+4, 5*k+4)*binomial(2*k, k)/(k+1));
(PARI) my(N=40, x='x+O('x^N)); Vec(2/((1-x)^2*((1-x)^3+sqrt((1-x)^6-4*x^5*(1-x)))))

Crossrefs

Cf. A086615, A360045, A360046.

Cf. A000108.

Sequence in context: A264925 A049016 A139761 * A275935 A137360 A195760

Adjacent sequences: A360044 A360045 A360046 * A360048 A360049 A360050

Keyword

nonn

Author

Seiichi Manyama, Jan 23 2023

Status

approved