A360100 a(n) = Sum_{k=0..n} binomial(n+2*k-1,n-k) * Catalan(k).
1, 1, 5, 23, 111, 562, 2952, 15948, 88076, 495077, 2823293, 16295020, 95007654, 558765743, 3310999269, 19748462718, 118471172054, 714355994997, 4327148812557, 26319195869861, 160677354596769, 984236344800234, 6047526697800992, 37262944840704171
Offset: 0
Keywords
Crossrefs
Programs
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Maple
A360100 := proc(n) add(binomial(n+2*k-1,n-k)*A000108(k),k=0..n) ; end proc: seq(A360100(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
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Mathematica
m = 24; A[_] = 0; Do[A[x_] = 1 + x A[x]^2/(1 - x)^3 + O[x]^m // Normal, {m}]; CoefficientList[A[x], x] (* Jean-François Alcover, Aug 16 2023 *) -
PARI
a(n) = sum(k=0, n, binomial(n+2*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)^3)))
Formula
G.f. A(x) satisfies A(x) = 1 + x * A(x)^2 / (1-x)^3.
G.f.: c(x/(1-x)^3), where c(x) is the g.f. of A000108.
a(n) ~ sqrt(-2 + (35 - 3*sqrt(129))^(1/3) + (35 + 3*sqrt(129))^(1/3)) * (((7 + (262 - 6*sqrt(129))^(1/3) + (2*(131 + 3*sqrt(129)))^(1/3))/3)^n / (sqrt(2*Pi) * n^(3/2))). - Vaclav Kotesovec, Feb 18 2023
D-finite with recurrence (n+1)*a(n) +(-8*n+5)*a(n-1) +(10*n-27)*a(n-2) +(-4*n+17)*a(n-3) +(n-6)*a(n-4)=0. - R. J. Mathar, Mar 12 2023