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A186925 Coefficient of x^n in (1+n*x+x^2)^n. 10
1, 1, 6, 45, 454, 5775, 88796, 1602447, 33213510, 777665691, 20302315252, 584774029983, 18422140045596, 630132567760345, 23257790717110392, 921362075184792825, 38994274473840538182, 1755943506127367745795 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..100

FORMULA

a(n) = [x^n](1+n*x+x^2)^n.

a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, n-2*k)*n^(n-2*k).

a(n) ~ BesselI(0,2) * n^n. - Vaclav Kotesovec, Apr 17 2014

a(n) = GegenbauerPoly(n,-n,-n/2). - Emanuele Munarini, Oct 20 2016

From Ilya Gutkovskiy, Sep 20 2017: (Start)

a(n) = [x^n] 1/sqrt((1 + 2*x - n*x)*(1 - 2*x - n*x)).

a(n) = n! * [x^n] exp(n*x)*BesselI(0,2*x). (End)

MATHEMATICA

Flatten[{1, Table[Sum[Binomial[n, k]*Binomial[n-k, n-2*k]*n^(n-2*k), {k, 0, Floor[n/2]}], {n, 1, 20}]}] (* Vaclav Kotesovec, Apr 17 2014 *)

Table[GegenbauerC[n, -n, -n/2] + KroneckerDelta[n, 0], {n, 0, 100}] (* Emanuele Munarini, Oct 20 2016 *)

PROG

(Maxima) a(n):=coeff(expand((1+n*x+x^2)^n), x, n);

(Maxima) makelist(ultraspherical(n, -n, -n/2), n, 0, 12); /* Emanuele Munarini, Oct 20 2016 */

makelist(a(n), n, 0, 20);

(MAGMA) P<x>:=PolynomialRing(Integers()); [ Coefficients((1+n*x+x^2)^n)[n+1]: n in [0..22] ]; // Klaus Brockhaus, Mar 02 2011

CROSSREFS

Cf. A092366, A187018, A187019, A187021, A070910.

Sequence in context: A239910 A228194 A084064 * A294642 A109516 A245493

Adjacent sequences:  A186922 A186923 A186924 * A186926 A186927 A186928

KEYWORD

nonn,easy

AUTHOR

Emanuele Munarini, Mar 02 2011

STATUS

approved

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Last modified December 12 20:04 EST 2017. Contains 295954 sequences.