A190738 Central coefficients of the Riordan matrix A104259.

1, 4, 27, 212, 1785, 15630, 140287, 1280592, 11833389, 110360150, 1036670272, 9794291556, 92972640761, 886023463122, 8471878678545, 81236546627920, 780898417097733, 7522708492892214, 72607180401922894, 701969331508141900, 6796919869909393140

Offset: 0

Author

Emanuele Munarini, May 18 2011

Keywords

• nonn

Crossrefs

Cf. A104259.

Programs

Mathematica

Table[Sum[Binomial[2n,n+k]Binomial[n+2k,k](n+1)/(n+k+1),{k,0,n}],{n,0,20}]

Maxima

makelist(sum(binomial(2*n,n+k)*binomial(n+2*k,k)*(n+1)/(n+k+1),k,0,n),n,0,20);

Formula

a(n) = T(2*n,n), where T(n,k) = A104259(n,k).

a(n) = sum(k=0..n, binomial(2*n,n+k)*binomial(n+2*k,k)*(n+1)/(n+k+1)).

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

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