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A358518
a(n) = Sum_{k=0..n} binomial(n+3*k+3,n-k) * Catalan(k).
2
1, 5, 20, 85, 405, 2116, 11766, 68237, 407789, 2492553, 15506942, 97859544, 624880895, 4029896310, 26209648212, 171711104853, 1132143259711, 7506530891217, 50019287312324, 334784759816729, 2249720564735567, 15172573979205166, 102662981205576494
OFFSET
0,2
FORMULA
a(n) = binomial(n+3,3) + Sum_{k=0..n-1} a(k) * a(n-k-1).
G.f. A(x) satisfies A(x) = 1/(1-x)^4 + x * A(x)^2.
G.f.: 2 / ( (1-x)^4 * (1 + sqrt( 1 - 4*x/(1-x)^4 )) ).
D-finite with recurrence (n+1)*a(n) +(-9*n+2)*a(n-1) +2*(7*n-4)*a(n-2) +10*(-n+2)*a(n-3) +5*(n-3)*a(n-4) +(-n+4)*a(n-5)=0. - R. J. Mathar, Jan 25 2023
PROG
(PARI) a(n) = sum(k=0, n, binomial(n+3*k+3, n-k)*binomial(2*k, k)/(k+1));
(PARI) my(N=30, x='x+O('x^N)); Vec(2/((1-x)^4*(1+sqrt(1-4*x/(1-x)^4))))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jan 23 2023
STATUS
approved