A082759 a(n) = Sum_{k = 0..n} binomial(n,k)*trinomial(n,k), where trinomial(n,k) = trinomial coefficients.

1, 2, 8, 35, 160, 752, 3599, 17446, 85376, 420884, 2087008, 10398016, 52010479, 261021854, 1313707256, 6628095035, 33512880640, 169768235840, 861450392708, 4377796514152, 22277498220160, 113502759811000, 578931209245760, 2955873376166144, 15105883318474991

Offset

0,2

Comments

Central coefficients of A115990. - Paul Barry, Feb 25 2011

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

a(n) = Sum_{k = 0..n} C(n+k, n-k)*C(n, k). - Benoit Cloitre, Jun 20 2003

2*n*(2*n - 1)*(38*n - 53)*a(n) + ( - 760*n^3 + 1820*n^2 - 1252*n + 252)*a(n - 1) - 8*(n - 1)*(19*n^2 - 36*n + 9)*a(n - 2) - 3*(38*n - 15)*(n - 1)*(n - 2)*a(n - 3) = 0. - Vladeta Jovovic, Jul 15 2004

a(n) = Sum_{k = 0..n} C(2*n - k, k)*C(n, k). - Paul Barry, Jan 20 2005

a(n) ~ c * d^n / sqrt(Pi*n), where d = 5.21913624874158651... = (((1261 + 57*sqrt(57))^(2/3) + 112 + 10*(1261 + 57*sqrt(57))^(1/3))/(6*(1261 + 57*sqrt(57))^(1/3))) is the real root of the equation 4*d^3 - 20*d^2 - 4*d - 3 = 0 and c = 0.79036380822702870439029... = 1/114*sqrt(57)*sqrt((9747 + 57*sqrt(57))^(1/3)*(2*(9747 + 57*sqrt(57))^(2/3) + 912 + 57*(9747 + 57*sqrt(57))^(1/3)))/((9747 + 57*sqrt(57))^(1/3)) is the positive real root of the equation 1216*c^6 - 912*c^4 + 100*c^2 - 3 = 0. - Vaclav Kotesovec, Oct 24 2012 (updated Oct 16 2016, following a suggestion of Michael Somos)

G.f.: A(x) = x*B'(x)/B(x), where B(x) satisfies B(x) = x*(1 + 2*B(x) + 2*B(x)^2 + B(x)^3). - Vladimir Kruchinin, Jan 14 2015

a(n) = Sum_{k = 0..n} (-1)^k*C(n, k)*C(3*n - 2*k, n - k). - Peter Bala, Jul 13 2016

G.f. y = A(x) satisfies 0 = 1 + y*(3-2*x) + y^3*(-4+20*x+4*x^2+3*x^3). - Michael Somos, Oct 15 2016

From Peter Bala, Jan 09 2022: (Start)

a(n) = [x^n] (1 + 2*x + 2*x^2 + x^3)^n.

The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. (End)

a(n) = hypergeom([-n, -n, n + 1], [1/2, 1], 1/4). - Peter Luschny, Jan 04 2025

Example

G.f. = 1 + 2*x + 8*x^2 + 35*x^3 + 160*x^4 + 752*x^5 + 3599*x^6 + 17446*x^7 + ...

Maple

a := n -> hypergeom([-n, -n, n + 1], [1/2, 1], 1/4):
seq(simplify(a(n)), n = 0..24);  # Peter Luschny, Jan 04 2025

Mathematica

Table[Sum[Binomial[2 n - k, k] Binomial[n, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 24 2012; typo fixed by Vincenzo Librandi, May 07 2013 *)

Prog

(PARI) a(n)=sum(k=0, n, binomial(n+k, n-k)*binomial(n, k))

Crossrefs

Cf. A037011, A106228, A115990.

Sequence in context: A037618 A326294 A184786 * A243204 A279013 A137265

Adjacent sequences: A082756 A082757 A082758 * A082760 A082761 A082762

Keyword

nonn,easy

Author

Emanuele Munarini, May 21 2003

Status

approved