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%I #47 Sep 08 2022 08:45:56
%S 1,2,13,136,1921,33876,712909,17383584,481003009,14869654300,
%T 507406003501,18928740714192,765897591633409,33392080668673832,
%U 1559976990077534253,77717020110946293376,4111810085670587224065,230190619432401207833004,13591965974806603671569101
%N Coefficient of x^n in (1 + (n+1)*x + n*x^2)^n.
%H Seiichi Manyama, <a href="/A187021/b187021.txt">Table of n, a(n) for n = 0..381</a> (terms 0..100 from Vincenzo Librandi)
%H Paul Barry and Aoife Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry2/barry190r.html">Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths</a>, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - _N. J. A. Sloane_, Oct 08 2012
%F a(n) = [x^n] (1 + (n+1)*x + n*x^2)^n.
%F a(n) = n^(n/2)*GegenbauerPoly(n,-n,-(n+1)/(2*sqrt(n)). - _Emanuele Munarini_, Oct 20 2016
%F a(n) = Sum_{k=0..n} binomial(n,k)^2 * n^k. - _Paul D. Hanna_, Mar 29 2011
%F a(n) ~ n^(n-1/4) * exp(2*sqrt(n)-1) / (2*sqrt(Pi)). - _Vaclav Kotesovec_, Apr 17 2014
%F a(n) = n! * [x^n] exp((n + 1)*x) * BesselI(0,2*sqrt(n)*x). - _Ilya Gutkovskiy_, May 31 2020
%F a(n) = hypergeom([-n, -n], [1], n). - _Peter Luschny_, Dec 22 2020
%p A187021:= n -> simplify( n^(n/2)*GegenbauerC(n, -n, -(n+1)/(2*sqrt(n))) );
%p 1, seq(A187021(n), n = 1..30); # _G. C. Greubel_, May 31 2020
%p a := n -> hypergeom([-n, -n], [1], n):
%p seq(simplify(a(n)), n=0..18); # _Peter Luschny_, Dec 22 2020
%t Flatten[{1,Table[Sum[Binomial[n,k]^2*n^k,{k,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, Apr 17 2014 *)
%t Table[If[n==0, 1, Simplify[n^(n/2)*GegenbauerC[n, -n, -(n+1)/(2 Sqrt[n])]]], {n, 0, 30}] (* _Emanuele Munarini_, Oct 20 2016 *)
%o (PARI) {a(n)=sum(k=0,n,binomial(n,k)^2*n^k)} \\ _Paul D. Hanna_, Mar 29 2011
%o (Maxima) a(n):=coeff(expand((1+(n+1)*x+n*x^2)^n),x,n);
%o makelist(a(n),n,0,20);
%o (Magma) P<x>:=PolynomialRing(Integers()); [ Coefficients((1+(n+1)*x+n*x^2)^n)[n+1]: n in [0..22] ]; // _Klaus Brockhaus_, Mar 03 2011
%o (Sage) [1]+[ n^(n/2)*gegenbauer(n, -n, -(n+1)/(2*sqrt(n))) for n in (1..30)] # _G. C. Greubel_, May 31 2020
%Y Main diagonal of A307883.
%Y Cf. A092366, A186925, A187018, A187019, A241247, A234971.
%K nonn
%O 0,2
%A _Emanuele Munarini_, Mar 02 2011
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