cp's OEIS Frontend

A360100 a(n) = Sum_{k=0..n} binomial(n+2*k-1,n-k) * Catalan(k).

%I A360100 #21 Aug 16 2023 08:21:59
%S A360100 1,1,5,23,111,562,2952,15948,88076,495077,2823293,16295020,95007654,
%T A360100 558765743,3310999269,19748462718,118471172054,714355994997,
%U A360100 4327148812557,26319195869861,160677354596769,984236344800234,6047526697800992,37262944840704171
%N A360100 a(n) = Sum_{k=0..n} binomial(n+2*k-1,n-k) * Catalan(k).
%F A360100 G.f. A(x) satisfies A(x) = 1 + x * A(x)^2 / (1-x)^3.
%F A360100 G.f.: c(x/(1-x)^3), where c(x) is the g.f. of A000108.
%F A360100 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
%F A360100 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
%p A360100 A360100 := proc(n)
%p A360100     add(binomial(n+2*k-1,n-k)*A000108(k),k=0..n) ;
%p A360100 end proc:
%p A360100 seq(A360100(n),n=0..70) ; # _R. J. Mathar_, Mar 12 2023
%t A360100 m = 24;
%t A360100 A[_] = 0;
%t A360100 Do[A[x_] = 1 + x A[x]^2/(1 - x)^3 + O[x]^m // Normal, {m}];
%t A360100 CoefficientList[A[x], x] (* _Jean-François Alcover_, Aug 16 2023 *)
%o A360100 (PARI) a(n) = sum(k=0, n, binomial(n+2*k-1, n-k)*binomial(2*k, k)/(k+1));
%o A360100 (PARI) my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x/(1-x)^3)))
%Y A360100 Partial sums are A360102.
%Y A360100 Cf. A162481, A258973.
%Y A360100 Cf. A002212, A006319, A162475, A360101.
%Y A360100 Cf. A000108.
%K A360100 nonn
%O A360100 0,3
%A A360100 _Seiichi Manyama_, Jan 25 2023