A360051 a(n) = Sum_{k=0..floor(n/5)} (-1)^k * binomial(n+4,5*k+4) * Catalan(k).

1, 5, 15, 35, 70, 125, 200, 275, 275, 0, -999, -3610, -9380, -20570, -39440, -65251, -85695, -56435, 141735, 781770, 2413128, 5999325, 12921350, 24387900, 39098925, 46638744, 11740695, -158571665, -674961760, -1956733020, -4724183860, -9957286550, -18316004575

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))) ).

Prog

(PARI) a(n) = sum(k=0, n\5, (-1)^k*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. A360048, A360049, A360050.

Cf. A000108.

Sequence in context: A292103 A373463 A290447 * A000750 A289389 A008487

Adjacent sequences: A360048 A360049 A360050 * A360052 A360053 A360054

Keyword

sign

Author

Seiichi Manyama, Jan 23 2023

Status

approved