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

%I A005191 M3891 #130 Mar 19 2025 08:03:35
%S A005191 1,1,5,19,85,381,1751,8135,38165,180325,856945,4091495,19611175,
%T A005191 94309099,454805755,2198649549,10651488789,51698642405,251345549849,
%U A005191 1223798004815,5966636799745,29125608152345,142330448514875,696235630761115,3408895901222375
%N A005191 Central pentanomial coefficients: largest coefficient of (1 + x + ... + x^4)^n.
%C A005191 Coefficient of x^n in ((1-x^10)/((1-x^5)(1-x^2)(1-x)))^n. - _Michael Somos_, Sep 24 2003
%C A005191 Note that n divides a(n+1) - a(n). - _T. D. Noe_, Mar 16 2005
%C A005191 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
%C A005191 Number of n-step 1-D walks ending at the origin with steps of size 0, 1 or 2. - _David Scambler_, Apr 09 2012
%C A005191 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
%C A005191 Let f(m) = ceiling((q+log(q))/log(25)), where q = -log(log(5)/(2*m^2*Pi)) then f(a(n)) = n, for n > 0. - _Miko Labalan_, Oct 07 2024
%D A005191 Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 603-604.
%D A005191 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005191 Alois P. Heinz, <a href="/A005191/b005191.txt">Table of n, a(n) for n = 0..1000</a> (first 201 terms from T. D. Noe)
%H A005191 Armen G. Bagdasaryan and Ovidiu Bagdasar, <a href="https://doi.org/10.1016/j.endm.2018.05.012">On some results concerning generalized arithmetic triangles</a>, Electronic Notes in Discrete Mathematics (2018) Vol. 67, 71-77.
%H A005191 V. E. Hoggatt, Jr. and M. Bicknell, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/7-4/hoggatt-a.pdf">Diagonal sums of generalized Pascal triangles</a>, Fib. Quart., 7 (1969), 341-358, 393.
%H A005191 Zagros Lalo, <a href="/A005191/a005191_2.pdf">Formula for Central terms in triangle A035343 ((1 + x + x^2 + x^3 + x^4)^n).</a>
%H A005191 Lyle E. Muller and Michelle Rudolph-Lilith, <a href="http://dx.doi.org/10.1016/j.disc.2015.04.001">On a link between Dirichlet kernels and central multinomial coefficients</a>, Discrete Mathematics, Volume 338, Issue 9, 6 September 2015, Pages 1567-1572.
%H A005191 Project Euler, <a href="https://projecteuler.net/problem=588">Quintinomial coefficients, Problem 588</a>
%H A005191 M. Rudolph-Lilith and L. E. Muller, <a href="http://arxiv.org/abs/1403.5942">On an explicit representation of central (2k+1)-nomial coefficients</a>, arXiv preprint arXiv:1403.5942 [math.CO], 2014.
%H A005191 <a href="/index/Ce#cfn">Index entries for sequences of k-nomial coefficients</a>
%F A005191 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
%F A005191 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]
%F A005191 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
%F A005191 a(n) ~ 5^n/(2*sqrt(Pi*n)). - _Vaclav Kotesovec_, Aug 09 2013
%F A005191 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
%p A005191 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
%t A005191 Flatten[{1,Table[Coefficient[Expand[Sum[x^j,{j,0,4}]^n],x^(2*n)],{n,1,20}]}] (* _Vaclav Kotesovec_, Aug 09 2013 *)
%t A005191 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 *)
%t A005191 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 *)
%o A005191 (PARI) a(n)=if(n<0,0,polcoeff(((1-x^5)/(1-x)+x*O(x^(2*n)))^n,2*n))
%o A005191 (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))
%o A005191 (PARI) a(n) = sum(k=0,(2*n)\5,binomial(n,k)*binomial(-n,2*n-5*k)) /* _Max Alekseyev_ */
%o A005191 (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_ */
%o A005191 (PARI) a(n) = vecmax(Vec(Pol(vector(5,k,1))^n)); \\ _Michel Marcus_, Jan 29 2017
%o A005191 (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
%Y A005191 Cf. A035343, A349936 (bisection).
%Y A005191 Row 5 of A077042. Column 2 of A201551 (?)
%K A005191 nonn
%O A005191 0,3
%A A005191 _N. J. A. Sloane_