A360101 a(n) = Sum_{k=0..n} binomial(n+4*k-1,n-k) * Catalan(k).
1, 1, 7, 40, 234, 1432, 9078, 59113, 393125, 2659233, 18240801, 126588424, 887221916, 6271153060, 44652824248, 319990906290, 2306133322704, 16703784324239, 121534039921585, 887845073567240, 6509750423778460, 47888814944642434, 353362258550740732
Offset: 0
Keywords
Programs
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Maple
A360101 := proc(n) add(binomial(n+4*k-1,n-k)*A000108(k),k=0..n) ; end proc: seq(A360101(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
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Mathematica
m = 23; A[_] = 0; Do[A[x_] = 1 + x A[x]^2/(1 - x)^5 + O[x]^m // Normal, {m}]; CoefficientList[A[x], x] (* Jean-François Alcover, Aug 16 2023 *) -
PARI
a(n) = sum(k=0, n, binomial(n+4*k-1, n-k)*binomial(2*k, k)/(k+1));
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PARI
my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x/(1-x)^5)))
Formula
G.f. A(x) satisfies A(x) = 1 + x * A(x)^2 / (1-x)^5.
G.f.: c(x/(1-x)^5), where c(x) is the g.f. of A000108.
D-finite with recurrence (n+1)*a(n) +(-10*n+7)*a(n-1) +(19*n-56)*a(n-2) +10*(-2*n+9)*a(n-3) +5*(3*n-19)*a(n-4) +(-6*n+49)*a(n-5) +(n-10)*a(n-6)=0. - R. J. Mathar, Mar 12 2023