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A055834 T(2n,n), where T is the array in A055830. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 2 17:25 EST 2022. Contains 358510 sequences. (Running on oeis4.)