%I #62 Dec 11 2024 21:50:32

%S 1,1,4,18,85,413,2044,10248,51876,264550,1357070,6994780,36196706,

%T 187938842,978599560,5108177816,26721644973,140050505085,735254208670,

%U 3865837887450,20353393741065,107290306033845,566194674179160,2990958274811520,15814562990604300,83690040760923168

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

%H Vincenzo Librandi, <a href="/A055834/b055834.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..n} binomial(n+k-1,n)*binomial(k,n-k). - _Max Alekseyev_, Jun 17 2007

%F 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

%F a(n) ~ 27^n/5^n*sqrt(2/(15*Pi*n)). - _Vaclav Kotesovec_, Nov 19 2012

%F a(n) = A055835(n)/3 for n>=1. - _Philippe Deléham_, Jan 25 2014

%F 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

%p seq( add(binomial(n+k-1,n)*binomial(k,n-k), k=0..n), n=0..30); # _G. C. Greubel_, Jan 21 2020

%t Table[Sum[Binomial[n+k-1,n]Binomial[k,n-k],{k,0,n}],{n,0,30}] (* _Harvey P. Dale_, Oct 03 2011 *)

%o (PARI) a(n) = sum(k=0,n,binomial(n+k-1,n)*binomial(k,n-k)); \\ _Joerg Arndt_, May 06 2013

%o (Maxima)

%o b(n):= sum(binomial(n+k, k)*binomial(k, n-k), k,ceiling(n/2),n)/(n+1);

%o B(x):= sum(b(i)*x^(i),i,0,30);

%o 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 */

%o (Magma) [&+[(Binomial(n+k-1, n)*Binomial(k, n-k)): k in [0..n]]: n in [0..30]]; // _Vincenzo Librandi_, Sep 21 2015

%o (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

%o (GAP) List([0..30], n-> Sum([0..n], k-> Binomial(n+k-1,n)*Binomial(k,n-k)) ); # _G. C. Greubel_, Jan 21 2020

%Y Cf. A001002, A055830, A055835.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, May 28 2000