A360051 - OEIS

%I #13 Jan 24 2023 02:56:27

%S 1,5,15,35,70,125,200,275,275,0,-999,-3610,-9380,-20570,-39440,-65251,

%T -85695,-56435,141735,781770,2413128,5999325,12921350,24387900,

%U 39098925,46638744,11740695,-158571665,-674961760,-1956733020,-4724183860,-9957286550,-18316004575

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

%H Seiichi Manyama, <a href="/A360051/b360051.txt">Table of n, a(n) for n = 0..1000</a>

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

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

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

%o (PARI) a(n) = sum(k=0, n\5, (-1)^k*binomial(n+4, 5*k+4)*binomial(2*k, k)/(k+1));

%o (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)))))

%Y Cf. A360048, A360049, A360050.

%Y Cf. A000108.

%K sign

%O 0,2

%A _Seiichi Manyama_, Jan 23 2023