A187021 Coefficient of x^n in (1 + (n+1)*x + n*x^2)^n.
1, 2, 13, 136, 1921, 33876, 712909, 17383584, 481003009, 14869654300, 507406003501, 18928740714192, 765897591633409, 33392080668673832, 1559976990077534253, 77717020110946293376, 4111810085670587224065, 230190619432401207833004, 13591965974806603671569101
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..381 (terms 0..100 from Vincenzo Librandi)
- Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - _N. J. A. Sloane_, Oct 08 2012
Programs
-
Magma
P
:=PolynomialRing(Integers()); [ Coefficients((1+(n+1)*x+n*x^2)^n)[n+1]: n in [0..22] ]; // Klaus Brockhaus, Mar 03 2011 -
Maple
A187021:= n -> simplify( n^(n/2)*GegenbauerC(n, -n, -(n+1)/(2*sqrt(n))) ); 1, seq(A187021(n), n = 1..30); # G. C. Greubel, May 31 2020 a := n -> hypergeom([-n, -n], [1], n): seq(simplify(a(n)), n=0..18); # Peter Luschny, Dec 22 2020
-
Mathematica
Flatten[{1,Table[Sum[Binomial[n,k]^2*n^k,{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Apr 17 2014 *) 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 *) -
Maxima
a(n):=coeff(expand((1+(n+1)*x+n*x^2)^n),x,n); makelist(a(n),n,0,20);
-
PARI
{a(n)=sum(k=0,n,binomial(n,k)^2*n^k)} \\ Paul D. Hanna, Mar 29 2011 -
Sage
[1]+[ n^(n/2)*gegenbauer(n, -n, -(n+1)/(2*sqrt(n))) for n in (1..30)] # G. C. Greubel, May 31 2020
Formula
a(n) = [x^n] (1 + (n+1)*x + n*x^2)^n.
a(n) = n^(n/2)*GegenbauerPoly(n,-n,-(n+1)/(2*sqrt(n))). - Emanuele Munarini, Oct 20 2016
a(n) = Sum_{k=0..n} binomial(n,k)^2 * n^k. - Paul D. Hanna, Mar 29 2011
a(n) ~ n^(n-1/4) * exp(2*sqrt(n)-1) / (2*sqrt(Pi)). - Vaclav Kotesovec, Apr 17 2014
a(n) = n! * [x^n] exp((n + 1)*x) * BesselI(0,2*sqrt(n)*x). - Ilya Gutkovskiy, May 31 2020
a(n) = hypergeom([-n, -n], [1], n). - Peter Luschny, Dec 22 2020