A359643 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*k,k).

1, 5, 37, 317, 2885, 27105, 259765, 2523813, 24768069, 244941833, 2437083697, 24367722725, 244639635749, 2464477467769, 24899468129405, 252202062544617, 2560119328830725, 26038134699958233, 265278657849511561, 2706809063101138409, 27657194997231516145, 282941098708193905485

Offset: 0

Author

Vaclav Kotesovec, Jan 09 2023

Keywords

• nonn

Comments

In general, for m>1, Sum_{k=0..n} binomial(n,k) * binomial(m*k,k) ~ sqrt((m + (1 - 1/m)^(m-1))/(m-1)) * (1 + m^m/(m-1)^(m-1))^n / sqrt(2*Pi*n).

Links

• Robert Israel, Table of n, a(n) for n = 0..977

Crossrefs

Cf. A026375, A188686.

Cf. A156887, A346646, A346664.

Programs

Maple

A359643 := proc(n)
    hypergeom([-n,1/4,1/2,3/4],[1/3,2/3,1],-256/27) ;
    simplify(%) ;
end proc:
seq(A359643(n),n=0..40) ; # R. J. Mathar, Jan 10 2023

Mathematica

Table[Sum[Binomial[n, k]*Binomial[4*k, k], {k, 0, n}], {n, 0, 20}]

PARI

a(n) = sum(k=0, n, binomial(n,k) * binomial(4*k,k)); \\ Michel Marcus, Jan 09 2023

Formula

a(n) ~ 283^(n + 1/2) / (2^(7/2) * sqrt(Pi*n) * 3^(3*n + 1/2)).

Conjecture D-finite with recurrence +81*n*(3*n-1)*(3*n-2)*a(n) +3*(243*n^3-8433*n^2+14984*n-7064)*a(n-1) +2*(-58607*n^3+297306*n^2-491401*n+269124)*a(n-2) +6*(n-2)*(56663*n^2-237722*n+252221)*a(n-3) -3*(n-2)*(n-3)*(111625*n-286402)*a(n-4) +110653*(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jan 09 2023

a(n) = 4F3( -n,1/4,1/2,3/4 ; 1/3, 2/3,1 ; -256/27). - R. J. Mathar, Jan 10 2023

a(n) = [x^n] (1 + 5*x + 6*x^2 + 4*x^3 + x^4)^n. - Ilya Gutkovskiy, Apr 17 2025