A055834 a(n) = T(2n,n), where T is the array in A055830.

1, 1, 4, 18, 85, 413, 2044, 10248, 51876, 264550, 1357070, 6994780, 36196706, 187938842, 978599560, 5108177816, 26721644973, 140050505085, 735254208670, 3865837887450, 20353393741065, 107290306033845, 566194674179160, 2990958274811520, 15814562990604300, 83690040760923168

Offset: 0

Author

Clark Kimberling, May 28 2000

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A001002, A055830, A055835.

Programs

GAP

List([0..30], n-> Sum([0..n], k-> Binomial(n+k-1,n)*Binomial(k,n-k)) ); # G. C. Greubel, Jan 21 2020

Magma

[&+[(Binomial(n+k-1, n)*Binomial(k, n-k)): k in [0..n]]: n in [0..30]]; // Vincenzo Librandi, Sep 21 2015

Maple

seq( add(binomial(n+k-1,n)*binomial(k,n-k), k=0..n), n=0..30); # G. C. Greubel, Jan 21 2020

Mathematica

Table[Sum[Binomial[n+k-1,n]Binomial[k,n-k],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Oct 03 2011 *)

Maxima

b(n):= sum(binomial(n+k, k)*binomial(k, n-k), k,ceiling(n/2),n)/(n+1);
B(x):= sum(b(i)*x^(i),i,0,30);
makelist(coeff(taylor(x*diff(B(x),x)-x*diff(B(x),x)/B(x)+B(x), x,0,20), x,n), n,0,20); /* Vladimir Kruchinin, Sep 21 2015 */

PARI

a(n) = sum(k=0,n,binomial(n+k-1,n)*binomial(k,n-k)); \\ Joerg Arndt, May 06 2013

Sage

[sum(binomial(n+k-1,n)*binomial(k,n-k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Jan 21 2020

Formula

a(n) = Sum_{k=0..n} binomial(n+k-1,n)*binomial(k,n-k). - Max Alekseyev, Jun 17 2007

Recurrence: 5*(n-1)*n*a(n) = 2*(n-1)*(11*n-3)*a(n-1) + 3*(3*n-5)*(3*n-4)*a(n-2). - Vaclav Kotesovec, Nov 19 2012

a(n) ~ 27^n/5^n*sqrt(2/(15*Pi*n)). - Vaclav Kotesovec, Nov 19 2012

a(n) = A055835(n)/3 for n>=1. - Philippe Deléham, Jan 25 2014

G.f.: x*B'(x)-x*B'(x)/B(x)+B(x), where B(x) is g.f. of A001002. - Vladimir Kruchinin, Sep 20 2015