A055834 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

0,3

Links

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

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

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 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(n+k-1, n)*binomial(k, n-k)); \\ Joerg Arndt, May 06 2013
(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 */
(Magma) [&+[(Binomial(n+k-1, n)*Binomial(k, n-k)): k in [0..n]]: n in [0..30]]; // Vincenzo Librandi, Sep 21 2015
(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
(GAP) List([0..30], n-> Sum([0..n], k-> Binomial(n+k-1, n)*Binomial(k, n-k)) ); # G. C. Greubel, Jan 21 2020

Crossrefs

Cf. A001002, A055830, A055835.

Sequence in context: A110166 A046885 A247637 * A079567 A030278 A318217

Adjacent sequences: A055831 A055832 A055833 * A055835 A055836 A055837

Keyword

nonn,easy

Author

Clark Kimberling, May 28 2000

Status

approved