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A262442 a(n) = Sum_{k=0..n}(binomial(n-1,n-k)*binomial(n+k-1,n-k)).

%I A262442 #23 Sep 08 2022 08:46:14
%S A262442 1,1,3,12,53,244,1152,5531,26875,131760,650492,3229368,16105344,
%T A262442 80624935,404913225,2039146908,10293657125,52071019600,263888886528,
%U A262442 1339540863092,6809667715812,34663102092960,176655038497000,901269559693104
%N A262442 a(n) = Sum_{k=0..n}(binomial(n-1,n-k)*binomial(n+k-1,n-k)).
%H A262442 Seiichi Manyama, <a href="/A262442/b262442.txt">Table of n, a(n) for n = 0..1000</a>
%F A262442 G.f.: 1+A'(x)*(x*A(x)-x^2)/A(x)^2, where A(x) is g.f. of A109081.
%F A262442 Recurrence: 2*(n-1)*(2*n - 1)*(38*n^2 - 162*n + 163)*a(n) = 2*(380*n^4 - 2380*n^3 + 5200*n^2 - 4676*n + 1431)*a(n-1) + 2*(n-2)*(76*n^3 - 362*n^2 + 502*n - 189)*a(n-2) + 3*(n-3)*(n-2)*(38*n^2 - 86*n + 39)*a(n-3). - _Vaclav Kotesovec_, Sep 23 2015
%F A262442 a(n) = n*hypergeom([1 - n, 1 - n, n + 1], [1, 3/2], 1/4) for n >= 1. - _Peter Luschny_, Mar 07 2022
%t A262442 Join[{1}, Table[Sum[Binomial[n-1, n-k] Binomial[ n+k-1, n-k], {k, n}], {n, 25}]] (* _Vincenzo Librandi_, Sep 23 2015 *)
%o A262442 (Maxima)
%o A262442 a(n):=sum(binomial(n, k)*binomial(n+k-2, n-k-1), k, 0, n-1)/n;
%o A262442 A(x):=sum(a(n)*x^n, n, 1, 30);
%o A262442 taylor(x*diff(A(x),x)/A(x)-x^2*diff(1/x-1/A(x),x),x,0,10);
%o A262442 (Magma) [&+[Binomial(n-1, n-k)*Binomial(n+k-1, n-k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Sep 23 2015
%o A262442 (PARI) a(n) = sum(k=0,n,(binomial(n-1,n-k)*binomial(n+k-1,n-k))) \\ _Anders Hellström_, Sep 23 2015
%Y A262442 Cf. A109081, A262440, A262441.
%K A262442 nonn
%O A262442 0,3
%A A262442 _Vladimir Kruchinin_, Sep 23 2015