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A005191 Central pentanomial coefficients: largest coefficient of (1 + x + ... + x^4)^n.
(Formerly M3891)
54
1, 1, 5, 19, 85, 381, 1751, 8135, 38165, 180325, 856945, 4091495, 19611175, 94309099, 454805755, 2198649549, 10651488789, 51698642405, 251345549849, 1223798004815, 5966636799745, 29125608152345, 142330448514875, 696235630761115 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Coefficient of x^n in ((1-x^10)/((1-x^5)(1-x^2)(1-x)))^n. - Michael Somos, Sep 24 2003

Note that n divides a(n+1) - a(n). - T. D. Noe, Mar 16 2005

Terms that are not a multiple of 5 have zero density, namely, there are fewer than n^(log(4)/log(5)) such terms among A005191(1..n). In particular, A005191(5k+2) and A005191(5k+4) are multiples of 5 for every k. - Max Alekseyev, Apr 25 2005

Number of n-step 1-D walks ending at the origin with steps of size 0, 1 or 2. - David Scambler, Apr 09 2012

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=0..200

V. E. Hoggatt, Jr. and M. Bicknell, Diagonal sums of generalized Pascal triangles, Fib. Quart., 7 (1969), 341-358, 393.

Lyle E. Muller and Michelle Rudolph-Lilith, On a link between Dirichlet kernels and central multinomial coefficients, Discrete Mathematics, Volume 338, Issue 9, 6 September 2015, Pages 1567-1572.

Project Euler, Quintinomial coefficients, Problem 588

M. Rudolph-Lilith, L. E. Muller, On an explicit representation of central (2k+1)-nomial coefficients, arXiv preprint arXiv:1403.5942, 2014

FORMULA

a(n) = Sum_{k=0..floor(2n/5)} binomial(n,k)*binomial(-n, 2n-5k); a(n) = (5^n + Sum_{j=1..2n-1} (sin(5j*Pi/(2n))/sin(j*Pi/(2n)))^n)/(2n) - 2. - Max Alekseyev, Mar 04 2005

Conjecture: 2*n*(2*n-1)*(3*n-4)*a(n) - (3*n-1)*(19*n^2-38*n+18)*a(n-1) - 5*(n-1)*(3*n-4)*(2*n-1)*a(n-2) + 25*(n-1)*(n-2)*(3*n-1)*a(n-3) = 0. [R. J. Mathar, Feb 21 2010] Proved using the Almkvist-Zeilberger algorithm in EKHAD. - Doron Zeilberger, Apr 02 2013

G.f.: sqrt((-5*x+2+2*sqrt(5*x^2-6*x+1))/(25*x^3-10*x^2-19*x+4))  - Mark van Hoeij, May 06 2013

a(n) ~ 5^n/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 09 2013

MATHEMATICA

Flatten[{1, Table[Coefficient[Expand[Sum[x^j, {j, 0, 4}]^n], x^(2*n)], {n, 1, 20}]}] (* Vaclav Kotesovec, Aug 09 2013 *)

PROG

(PARI) a(n)=if(n<0, 0, polcoeff(((1-x^5)/(1-x)+x*O(x^(2*n)))^n, 2*n))

(PARI) a(n)=if(n<0, 0, polcoeff(((1-x^10)/((1-x^5)*(1-x^2)*(1-x))+x*O(x^n))^n, n))

(PARI) a(n) = sum(k=0, (2*n)\5, binomial(n, k)*binomial(-n, 2*n-5*k)) /* Max Alekseyev */

(PARI) a(n) = round((5^n+sum(j=1, 2*n-1, (sin(5*Pi*j/2/n)/sin(Pi*j/2/n))^n))/2/n)-2 /* Max Alekseyev */

(PARI) a(n) = vecmax(Vec(Pol(vector(5, k, 1))^n)); \\ Michel Marcus, Jan 29 2017

CROSSREFS

Cf. A001405, A002426, A005190, A018901, A025012, A025013, A025014.

Sequence in context: A149794 A149795 A149796 * A275027 A147091 A149797

Adjacent sequences:  A005188 A005189 A005190 * A005192 A005193 A005194

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified September 22 00:25 EDT 2017. Contains 292326 sequences.