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A262440 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k-1,n-k).

%I A262440 #32 Apr 30 2024 04:51:55
%S A262440 1,1,5,22,101,476,2282,11075,54245,267592,1327580,6617128,33110090,
%T A262440 166215895,836761343,4222640822,21354409445,108193910000,549084400088,
%U A262440 2790744368660,14203023709276,72371208424880,369170645788840,1885051297844624
%N A262440 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k-1,n-k).
%H A262440 Seiichi Manyama, <a href="/A262440/b262440.txt">Table of n, a(n) for n = 0..1000</a>
%F A262440 G.f.: x*A'(x)/A(x), where A(x) is g.f. of A109081.
%F A262440 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 A262440 a(n) = n^2*hypergeom([1-n, 1-n, n+1], [3/2, 2], 1/4) for n >= 1. - _Peter Luschny_, Mar 06 2022
%F A262440 a(n) = [x^n] ( (1 - x + x^2) / (1 - x)^2 )^n. - _Seiichi Manyama_, Apr 29 2024
%F A262440 a(n) ~ sqrt((513 - 67*sqrt(57))^(1/3) + (513 + 67*sqrt(57))^(1/3)) * (10 + (1261 - 57*sqrt(57))^(1/3) + (1261 + 57*sqrt(57))^(1/3))^n / (19^(1/3) * sqrt(Pi*n) * 2^(n + 5/6) * 3^(n + 1/3)). - _Vaclav Kotesovec_, Apr 30 2024
%t A262440 Join[{1}, Table[Sum[ Binomial[n,k] Binomial[n+k-1, n-k], {k, n}], {n, 25}]] (* _Vincenzo Librandi_, Sep 23 2015 *)
%o A262440 (Maxima)
%o A262440 a(n):=sum(binomial(n,k)*binomial(n+k-2,n-k-1),k,0,n-1)/n;
%o A262440 A(x):=sum(a(n)*x^n,n,1,30);
%o A262440 taylor(diff(A(x),x)/A(x)*x,x,0,10);
%o A262440 (Magma) [&+[Binomial(n, k)*Binomial(n+k-1, n-k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Sep 13 2015
%o A262440 (PARI)  a(n)=sum(k=0,n,(binomial(n,k)*binomial(n+k-1,n-k))) \\ _Anders Hellström_, Sep 23 2015
%Y A262440 Cf. A109081, A246437, A262441, A262442.
%K A262440 nonn
%O A262440 0,3
%A A262440 _Vladimir Kruchinin_, Sep 23 2015