A055835 T(2n+1,n), where T is the array in A055830.

1, 3, 12, 54, 255, 1239, 6132, 30744, 155628, 793650, 4071210, 20984340, 108590118, 563816526, 2935798680, 15324533448, 80164934919, 420151515255, 2205762626010, 11597513662350, 61060181223195, 321870918101535

Offset: 0

Author

Clark Kimberling, May 28 2000

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A055830, A055834.

Programs

GAP

Concatenation([1], List([1..30], n-> 3*Sum([0..n], k-> Binomial(n+k-1, n) *Binomial(k, n-k) ))); # G. C. Greubel, Jan 21 2020

Magma

[1] cat [3*(&+[Binomial(n+k-1, n)*Binomial(k, n-k): k in [0..n]]): n in [1..30]]; // G. C. Greubel, Jan 21 2020

Maple

seq( `if`(n=0, 1, 3*add(binomial(n+k-1, n)*binomial(k, n-k), k=0..n)), n=0..30); # G. C. Greubel, Jan 21 2020

Mathematica

Table[If[n==0, 1, 3*Sum[Binomial[k, n-k]*Binomial[n+k-1, n], {k,0,n}]], {n,0,30}] (* G. C. Greubel, Jan 21 2020 *)

Maxima

a(n):=sum((sum(binomial(i,k)*binomial(i+1,n-i),i,k,n))*binomial(n,k), k,0,n); /* Vladimir Kruchinin, Mar 01 2014 */

PARI

a(n) = if(n==0, 1, 3*sum(k=0, n, binomial(n+k-1, n)*binomial(k, n-k)) ); \\ Joerg Arndt, Mar 01 2014

Sage

[1]+[3*sum(binomial(n+k-1,n)*binomial(k,n-k) for k in (0..n)) for n in (1..30)] # G. C. Greubel, Jan 21 2020

Formula

a(n) = 3*A055834(n) for n>=1. - Philippe Deléham, Jan 25 2014

a(n) = Sum_{k=0..n} Sum_{i=k..n} binomial(i,k)*binomial(i+1,n-i)*binomial(n,k). - Vladimir Kruchinin, Mar 01 2014