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A349768
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * binomial(2*k,k) / (k+1).
1
1, 3, 19, 173, 1881, 22655, 291775, 3940725, 55149025, 793387235, 11668476579, 174735112997, 2656296912361, 40897718776647, 636588467802679, 10002872642155085, 158483629611962025, 2529389028336106475, 40631849127696017275, 656509442594976984405, 10663184061320964941761
OFFSET
0,2
FORMULA
From Vaclav Kotesovec, Nov 29 2021: (Start)
D-finite recurrence: n*(n+1)*(2*n - 3)*a(n) = (2*n - 1)*(19*n^2 - 37*n + 12)*a(n-1) - (2*n - 3)*(19*n^2 - 39*n + 14)*a(n-2) + (n-3)*(n-2)*(2*n - 1)*a(n-3).
a(n) ~ sqrt(5) * phi^(6*n + 3) / (8*Pi*n^2), where phi = A001622 is the golden ratio. (End)
D-finite with recurrence n*(n+1)*a(n) +(n+1)*(n-4)*a(n-1) +2*(-171*n^2 +512*n -388)*a(n-2) +2*(9*n^2 +296*n -796)*a(n-3) +(341*n^2 -2425*n +4320)*a(n-4) -19*(n-4)*(n-5)*a(n-5)=0. - R. J. Mathar, Mar 02 2023
MAPLE
A349768 := proc(n)
hypergeom([1/2, -n, n+1], [1, 2], -4) ;
simplify(%) ;
end proc:
seq(A349768(n), n=0..20) ; # R. J. Mathar, Mar 02 2023
MATHEMATICA
Table[Sum[Binomial[n, k] Binomial[n + k, k] Binomial[2 k, k]/(k + 1), {k, 0, n}], {n, 0, 20}]
Table[HypergeometricPFQ[{1/2, -n, n + 1}, {1, 2}, -4], {n, 0, 20}]
PROG
(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(n+k, k)*binomial(2*k, k)/(k+1)); \\ Michel Marcus, Nov 29 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ilya Gutkovskiy, Nov 29 2021
STATUS
approved