A050152 a(n) = T(n,n+3), array T as in A050143.

0, 1, 7, 42, 242, 1375, 7773, 43876, 247684, 1399293, 7913955, 44812878, 254064726, 1442131899, 8195232633, 46621424520, 265490365448, 1513290869881, 8633347134975, 49293941140402, 281670686307130

Offset: 0

Author

Clark Kimberling, Dec 11 1999

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Milan Janjic, Two Enumerative Functions

Crossrefs

Cf. A050143.

Programs

Maple

a := n -> (-1)^n*(n+1)*(n/2)*hypergeom([-n,n+2], [3], 2);
seq(round(evalf(a(n),32)), n=0..20); # Peter Luschny, May 24 2014

Mathematica

Table[JacobiP[n-1, 1, 2, 3], {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Sep 12 2008 *)

Maxima

a(n):=sum(2^(i-2)*(-1)^(n-i)*binomial(n,n-i+2)*binomial(n+i-1,n-1),i,0,n+2); /* Vladimir Kruchinin, May 24 2014 */

Formula

G.f.: (x*sqrt(x^2-6*x+1)-x^2+3*x)/(-x^4+sqrt(x^2-6*x+1)*(x^3-3*x^2-3*x+1)+6*x^3-2*x^2+6*x-1)+1/(4*x)-1/(4*x^2). - Vladimir Kruchinin, May 24 2014

a(n) = Sum_{i=0..n+2} 2^(i-2)*(-1)^(n-i)*binomial(n,n-i+2)*binomial(n+i-1,n-1). - Vladimir Kruchinin, May 24 2014

a(n) ~ sqrt(48+34*sqrt(2)) * (3+2*sqrt(2))^n / (8*sqrt(Pi*n)). - Vaclav Kotesovec, May 24 2014

a(n) = (-1)^n*(n+1)*(n/2)*hypergeom([-n, n+2], [3], 2). - Peter Luschny, May 24 2014

n^2*(n+1)*a(n-1) = Sum_{k=0..n-1} (2*k^3+k^2+k)*binomial(n-1,k)*binomial(n+k,k) for all n > 0. This follows from the Zeilberger algorithm. - Zhi-Wei Sun, Aug 30 2014

a(n) = Sum_{k=0..n} (binomial(n,k)*binomial(2*n-k+1,n-k-1)). - Vladimir Kruchinin, Oct 26 2016

Extensions

Typo in Mathematica code fixed by Vincenzo Librandi, May 26 2013