A005191 Central pentanomial coefficients: largest coefficient of (1 + x + ... + x^4)^n. (Formerly M3891)

1, 1, 5, 19, 85, 381, 1751, 8135, 38165, 180325, 856945, 4091495, 19611175, 94309099, 454805755, 2198649549, 10651488789, 51698642405, 251345549849, 1223798004815, 5966636799745, 29125608152345, 142330448514875, 696235630761115, 3408895901222375

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

Number of compositions of 2n into exactly n nonnegative parts <= four. a(2) = 5: [4,0], [3,1], [2,2], [1,3], [0,4]. - Alois P. Heinz, Sep 13 2018

References

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 603-604.

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from T. D. Noe)

Armen G. Bagdasaryan, Ovidiu Bagdasar, On some results concerning generalized arithmetic triangles, Electronic Notes in Discrete Mathematics (2018) Vol. 67, 71-77.

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

Zagros Lalo, Formula for Central terms in triangle A035343 ((1 + x + x^2 + x^3 + x^4)^n).

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 [math.CO], 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

D-finite with recurrence: 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

a(n) = Sum_{i=0..n/2} Sum_{j=0..n} Sum_{q=n..2*n}(f); f=( n!/((q - n)!*(j - 2*q + 2*n)!*(i - 2*j + q)!*(j - 2*i)!*i!) ); f=0 for (j - 2*q + 2*n)<0 or (i - 2*j + q)<0 or (j - 2*i)<0. Also see formula in Links section. - Zagros Lalo, Sep 25 2018

Maple

seq(coeff(series(((1-x^10)/((1-x^5)*(1-x^2)*(1-x)))^n, x, n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Sep 26 2018

Mathematica

Flatten[{1, Table[Coefficient[Expand[Sum[x^j, {j, 0, 4}]^n], x^(2*n)], {n, 1, 20}]}] (* Vaclav Kotesovec, Aug 09 2013 *)
a[n_] := a[n] = Sum[n!/((q - n)!*(j - 2*q + 2*n)!*(i - 2*j + q)!*(j - 2*i)!*i!), {i, 0, n/2}, {j, 0, n}, {q, n, 2*n}]; Table[a[n], {n, 0, 29}] (* Zagros Lalo, Sep 25 2018 *)
CoefficientList[Series[Sqrt[(-5x+2+2Sqrt[5x^2-6x+1])/(25x^3-10x^2-19x+4)], {x, 0, 30}], x] (* Harvey P. Dale, Aug 04 2021 *)

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
(GAP) List([0..25], n->Sum([0..Int(2*n/5)], k->Binomial(n, k)*Binomial(-n, 2*n-5*k))); # Muniru A Asiru, Sep 26 2018

Crossrefs

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

Cf. A035343.

Sequence in context: A149795 A149796 A348410 * A324595 A275027 A364743

Adjacent sequences: A005188 A005189 A005190 * A005192 A005193 A005194

Keyword

nonn

Author

N. J. A. Sloane

Status

approved