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a(n) = Sum_{k=0..floor(n/4)} binomial(n+3,4*k+3) * Catalan(k).
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%I #17 Jan 25 2023 08:32:05

%S 1,4,10,20,36,64,120,240,497,1036,2158,4524,9625,20816,45560,100368,

%T 221915,492380,1097302,2457228,5526666,12474000,28233600,64061920,

%U 145704327,332174532,758977386,1737703780,3985847284,9157908736,21074460512,48569746368,112096071675

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

%H Seiichi Manyama, <a href="/A360046/b360046.txt">Table of n, a(n) for n = 0..1000</a>

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

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

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

%F 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

%o (PARI) a(n) = sum(k=0, n\4, binomial(n+3, 4*k+3)*binomial(2*k, k)/(k+1));

%o (PARI) my(N=40, x='x+O('x^N)); Vec(2/((1-x)^2*((1-x)^2+sqrt((1-x)^4-4*x^4))))

%Y Cf. A086615, A360045, A360047.

%Y Cf. A000108.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jan 23 2023