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
Examples
G.f. = 1 + 2*x + 8*x^2 + 35*x^3 + 160*x^4 + 752*x^5 + 3599*x^6 + 17446*x^7 + ...
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Programs
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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
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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 *) -
PARI
a(n)=sum(k=0,n,binomial(n+k,n-k)*binomial(n,k))
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
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