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

0,2

Links

Table of n, a(n) for 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

Mathematica

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

Prog

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

Crossrefs

Cf. A104259.

Sequence in context: A026005 A059391 A371786 * A361717 A275607 A319518

Adjacent sequences: A190735 A190736 A190737 * A190739 A190740 A190741

Keyword

nonn

Author

Emanuele Munarini, May 18 2011

Status

approved