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A382859 a(n) = Sum_{k=0..n} binomial(n,k) * binomial((n-1)*(k+1),n-k).

%I A382859 #17 Apr 09 2025 09:57:09
%S A382859 1,1,5,37,345,3851,49468,713931,11391985,198523495,3741919446,
%T A382859 75702725440,1633591960883,37404262517506,904734768056239,
%U A382859 23030071358784701,614912094171482849,17172036245893988575,500281954849350450946,15170753984617328108901
%N A382859 a(n) = Sum_{k=0..n} binomial(n,k) * binomial((n-1)*(k+1),n-k).
%H A382859 Vincenzo Librandi, <a href="/A382859/b382859.txt">Table of n, a(n) for n = 0..150</a>
%F A382859 a(n) = [x^n] (1 + x/(1-x)^n)^n.
%t A382859 Table[Sum[Binomial[n,k] * Binomial[(n-1)*(k+1),n-k], {k,0,n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 07 2025 *)
%o A382859 (PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial((n-1)*(k+1), n-k));
%o A382859 (Magma) [&+[Binomial(n,k) * Binomial((n-1)*(k+1),n-k): k in [0..n]]: n in [0..21]]; // _Vincenzo Librandi_, Apr 09 2025
%Y A382859 Main diagonal of A381425.
%Y A382859 Cf. A121673, A121674, A121675.
%K A382859 nonn,easy
%O A382859 0,3
%A A382859 _Seiichi Manyama_, Apr 07 2025