|
%I #10 Jun 05 2022 03:14:24
%S 1,4,27,212,1785,15630,140287,1280592,11833389,110360150,1036670272,
%T 9794291556,92972640761,886023463122,8471878678545,81236546627920,
%U 780898417097733,7522708492892214,72607180401922894,701969331508141900,6796919869909393140
%N Central coefficients of the Riordan matrix A104259.
%F a(n) = T(2*n,n), where T(n,k) = A104259(n,k).
%F a(n) = sum(k=0..n, binomial(2*n,n+k)*binomial(n+2*k,k)*(n+1)/(n+k+1)).
%F Recurrence: 2*(n-1)*n*(2*n + 1)*(107*n^3 - 345*n^2 + 253*n + 12)*a(n) = (n-1)*(5029*n^5 - 16215*n^4 + 8284*n^3 + 13359*n^2 - 11675*n + 1974)*a(n-1) - 4*(2*n - 3)*(1498*n^5 - 4830*n^4 + 2899*n^3 + 3165*n^2 - 3254*n + 630)*a(n-2) + 100*(n-1)*(2*n - 5)*(2*n - 3)*(107*n^3 - 24*n^2 - 116*n + 27)*a(n-3). - _Vaclav Kotesovec_, Jul 05 2021
%F a(n) ~ sqrt(c) * d^n / sqrt(Pi*n), where d = 9.945658804810730213397409025621... is the real root of the equation -400 + 112*d - 47*d^2 + 4*d^3 = 0 and c = 0.3447849735035503206155951176700724872157... is the real root of the equation -125 - 173*c + 963*c^2 + 1712*c^3 = 0. - _Vaclav Kotesovec_, Jun 05 2022
%t Table[Sum[Binomial[2n,n+k]Binomial[n+2k,k](n+1)/(n+k+1),{k,0,n}],{n,0,20}]
%o (Maxima) makelist(sum(binomial(2*n,n+k)*binomial(n+2*k,k)*(n+1)/(n+k+1),k,0,n),n,0,20);
%Y Cf. A104259.
%K nonn
%O 0,2
%A _Emanuele Munarini_, May 18 2011
|
