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 A187018 Coefficient of x^n in (1 + x + n*x^2)^n. 11
 1, 1, 5, 19, 145, 851, 7741, 58605, 600769, 5420035, 61026901, 628076153, 7648488145, 87388647373, 1138801242125, 14182492489651, 196218339243777, 2628971539313875, 38377805385510181, 547815690902283225, 8395817775835635601, 126725586542235932329 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..590 (terms 0..122 from Vincenzo Librandi) FORMULA a(n) = [x^n] (1 + x + n*x^2)^n. a(n) = n^(n/2)*GegenbauerPoly(n,-n,-1/(2*sqrt(n))). - Emanuele Munarini, Oct 20 2016 a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, n-2*k)*n^k. a(n) ~ 2^(n-1/2) * exp(sqrt(n)/2-1/8) * n^(n/2-1/2) / sqrt(Pi). - Vaclav Kotesovec, Apr 17 2014 a(n) = [x^n] 1/sqrt(1 - 2*x - (4*n-1)*x^2). - Paul D. Hanna, Dec 12 2014 EXAMPLE G.f. = 1 + x + 5*x^2 + 19*x^3 + 145*x^4 + 851*x^5 + 7741*x^6 + 58605*x^7 + ... MATHEMATICA Flatten[{1, Table[Sum[Binomial[n, k]*Binomial[n-k, n-2*k]*n^k, {k, 0, Floor[n/2]}], {n, 1, 20}]}] (* Vaclav Kotesovec, Apr 17 2014 *) a[ n_] := SeriesCoefficient[ (1 + x + x^2)^n, {x, 0, n}]; (* Michael Somos, Dec 12 2014 *) Table[If[n == 0, 1, Simplify[n^(n/2) GegenbauerC[n, -n, -1/(2 Sqrt[n])]]], {n, 0, 12}] (* Emanuele Munarini, Oct 20 2016 *) PROG (Maxima) a(n):=coeff(expand((1+x+n*x^2)^n), x, n); makelist(a(n), n, 0, 20); (MAGMA) P:=PolynomialRing(Integers()); [ Coefficients((1+x+n*x^2)^n)[n+1]: n in [0..22] ]; // Klaus Brockhaus, Mar 03 2011 (PARI) {a(n)=polcoeff(1/sqrt(1 - 2*x - (4*n-1)*x^2 +x*O(x^n)), n)} for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Dec 12 2014 (PARI) {a(n) = polcoef((1+x+n*x^2)^n, n)} \\ Seiichi Manyama, May 01 2019 CROSSREFS Cf. A092366, A186925, A187019, A187021. Sequence in context: A209111 A297389 A228479 * A193287 A027269 A082790 Adjacent sequences:  A187015 A187016 A187017 * A187019 A187020 A187021 KEYWORD nonn,easy AUTHOR Emanuele Munarini, Mar 02 2011 STATUS approved

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Last modified October 13 20:38 EDT 2019. Contains 327981 sequences. (Running on oeis4.)