A360046 a(n) = Sum_{k=0..floor(n/4)} binomial(n+3,4*k+3) * Catalan(k).

1, 4, 10, 20, 36, 64, 120, 240, 497, 1036, 2158, 4524, 9625, 20816, 45560, 100368, 221915, 492380, 1097302, 2457228, 5526666, 12474000, 28233600, 64061920, 145704327, 332174532, 758977386, 1737703780, 3985847284, 9157908736, 21074460512, 48569746368, 112096071675

Offset

0,2

Links

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

Formula

a(n) = binomial(n+3,3) + Sum_{k=0..n-4} a(k) * a(n-k-4).

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

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

D-finite with recurrence (n+4)*a(n) +5*(-n-3)*a(n-1) +10*(n+2)*a(n-2) +10*(-n-1)*a(n-3) +(n+8)*a(n-4) +3*(n-1)*a(n-5)=0. - R. J. Mathar, Jan 25 2023

Prog

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

Crossrefs

Cf. A086615, A360045, A360047.

Cf. A000108.

Sequence in context: A131924 A143982 A000749 * A354696 A275934 A318338

Adjacent sequences: A360043 A360044 A360045 * A360047 A360048 A360049

Keyword

nonn

Author

Seiichi Manyama, Jan 23 2023

Status

approved