login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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<x>:=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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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