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A128079 a(n) = Sum_{k=0..n} A000984(k)*A001263(n+1,k+1), where A000984 is the central binomial coefficients and A001263 is the Narayana triangle. 1
1, 3, 13, 69, 411, 2633, 17739, 124029, 892327, 6567285, 49235715, 374841195, 2890994445, 22545855855, 177524073021, 1409591810133, 11275693221519, 90792020672429, 735367765159347, 5987665336600683, 48987680485918149 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

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

FORMULA

a(n) = Sum_{k=0..n} C(2k,k)*C(n,k)*C(n+1,k)/(k+1).

Recurrence: (n+1)*(n+2)*a(n) = (7*n^2+11*n+6)*a(n-1) + 3*(7*n^2-19*n+6)*a(n-2) - 27*(n-2)*(n-1)*a(n-3) . - Vaclav Kotesovec, Oct 20 2012

a(n) ~ 3^(2*n+7/2)/(8*Pi*n^2) . - Vaclav Kotesovec, Oct 20 2012

EXAMPLE

Illustrate a(n) = Sum_{k=0..n} A000984(k)*A001263(n+1,k+1) by:

a(2) = 1*(1) + 2*(3) + 6*(1) = 13;

a(3) = 1*(1) + 2*(6) + 6*(6) + 20*(1) = 69;

a(4) = 1*(1) + 2*(10)+ 6*(20)+ 20*(10)+ 70*(1) = 411.

The Narayana triangle A001263(n+1,k+1) = C(n,k)*C(n+1,k)/(k+1) begins:

1;

1, 1;

1, 3, 1;

1, 6, 6, 1;

1, 10, 20, 10, 1;

1, 15, 50, 50, 15, 1; ...

MATHEMATICA

Table[Sum[Binomial[2*k, k]*Binomial[n, k]*Binomial[n+1, k]/(k+1), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 20 2012 *)

PROG

(PARI) {a(n)=sum(k=0, n, binomial(2*k, k)*binomial(n, k)*binomial(n+1, k)/(k+1))}

CROSSREFS

Cf. A000984, A001263.

Sequence in context: A020107 A284718 A284719 * A074534 A153395 A243688

Adjacent sequences:  A128076 A128077 A128078 * A128080 A128081 A128082

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 23 2007

STATUS

approved

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Last modified April 18 08:37 EDT 2019. Contains 322209 sequences. (Running on oeis4.)