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%I #25 Sep 08 2022 08:46:14
%S 1,1,5,22,101,476,2282,11075,54245,267592,1327580,6617128,33110090,
%T 166215895,836761343,4222640822,21354409445,108193910000,549084400088,
%U 2790744368660,14203023709276,72371208424880,369170645788840,1885051297844624
%N a(n) = Sum_{k=0..n}(binomial(n,k)*binomial(n+k-1,n-k)).
%H Seiichi Manyama, <a href="/A262440/b262440.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: x*A'(x)/A(x), where A(x) is g.f. of A109081.
%F Recurrence: 2*n*(2*n-1)*(38*n^3 - 210*n^2 + 377*n - 219)*a(n) = 2*(380*n^5 - 2480*n^4 + 5998*n^3 - 6598*n^2 + 3219*n - 540)*a(n-1) + 2*(n-2)*(76*n^4 - 382*n^3 + 572*n^2 - 300*n + 45)*a(n-2) + 3*(n-3)*(n-2)*(38*n^3 - 96*n^2 + 71*n - 14)*a(n-3). - _Vaclav Kotesovec_, Sep 23 2015
%F a(n) = n^2*hypergeom([1-n, 1-n, n+1], [3/2, 2], 1/4) for n >= 1. - _Peter Luschny_, Mar 06 2022
%t Join[{1}, Table[Sum[ Binomial[n,k] Binomial[n+k-1, n-k], {k, n}], {n, 25}]] (* _Vincenzo Librandi_, Sep 23 2015 *)
%o (Maxima)
%o a(n):=sum(binomial(n,k)*binomial(n+k-2,n-k-1),k,0,n-1)/n;
%o A(x):=sum(a(n)*x^n,n,1,30);
%o taylor(diff(A(x),x)/A(x)*x,x,0,10);
%o (Magma) [&+[Binomial(n, k)*Binomial(n+k-1, n-k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Sep 13 2015
%o (PARI) a(n)=sum(k=0,n,(binomial(n,k)*binomial(n+k-1,n-k))) \\ _Anders Hellström_, Sep 23 2015
%Y Cf. A109081, A262441, A262442.
%K nonn
%O 0,3
%A _Vladimir Kruchinin_, Sep 23 2015
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