A360050 a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n+3,4*k+3) * Catalan(k).

1, 4, 10, 20, 34, 48, 48, 0, -163, -548, -1274, -2340, -3255, -2224, 5304, 28560, 82379, 182300, 322102, 410700, 133128, -1295264, -5440600, -14733680, -31384533, -52870668, -59633454, 11449780, 312532426, 1137823168, 2918752832, 5961965824, 9464314955

Offset

0,2

Links

Table of n, a(n) for n=0..32.

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) +(9*n-8)*a(n-4) +5*(-n+1)*a(n-5) =0. - R. J. Mathar, Jan 25 2023

Prog

(PARI) a(n) = sum(k=0, n\4, (-1)^k*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. A360048, A360049, A360051.

Cf. A000108.

Sequence in context: A008045 A008124 A019457 * A099589 A008141 A119651

Adjacent sequences: A360047 A360048 A360049 * A360051 A360052 A360053

Keyword

sign

Author

Seiichi Manyama, Jan 23 2023

Status

approved