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A025014 Central "nonomial" coefficient: largest coefficient of (1+x+...+x^8)^n.

%I A025014 #29 Mar 19 2025 08:06:00
%S A025014 1,1,9,61,489,3951,32661,273127,2306025,19610233,167729959,1441383219,
%T A025014 12434998005,107632809909,934263293679,8129320828911,70886845397481,
%U A025014 619288973447049,5419332253680705,47494787636620701,416800775902696839
%N A025014 Central "nonomial" coefficient: largest coefficient of (1+x+...+x^8)^n.
%C A025014 Generally, largest coefficient of (1+x+...+x^k)^n is asymptotic to (k+1)^n * sqrt(6/(k*(k+2)*Pi*n)). - _Vaclav Kotesovec_, Aug 09 2013
%H A025014 T. D. Noe, <a href="/A025014/b025014.txt">Table of n, a(n) for n=0..200</a>
%H A025014 Michelle Rudolph-Lilith and Lyle E. Muller, <a href="https://doi.org/10.1016/j.disc.2015.04.001">On a link between Dirichlet kernels and central multinomial coefficients</a>, Discrete Mathematics 338.9 (2015): 1567-1572.
%H A025014 <a href="/index/Ce#cfn">Index entries for sequences of k-nomial coefficients</a>
%F A025014 The Almkvist-Zeilberger algorithm in EKHAD establishes the following recurrence:
%F A025014 -6561*(4*n+17)*(4*n+13)*(5*n+24)*(5*n+19)*(5*n+14)*(5*n+23)*(n+4)*(n+3)*(n+2)*(n+1)*a(n)+1458*(5*n+24)*(5*n+19)*(4*n+17)*(5*n+9)*(4*n+9)*(5*n+18)*(2*n+9)*(n+4)*(
%F A025014 n+3)*(n+2)*a(n+1)+162*(5*n+24)*(5*n+14)*(4*n+13)*(5*n+23)*(n+4)*(n+3)*(1020*n^4+12291*n^3+53378*n^2+98617*n+65610)*a(n+2)-18*(4*n+17)*(4*n+9)*(5*n+19)*(2*n+9)*(5
%F A025014 *n+9)*(5*n+18)*(n+4)*(385*n^3+4158*n^2+14551*n+16610)*a(n+3)-(5*n+23)*(4*n+13)*(4*n+9)*(5*n+24)*(5*n+14)*(5*n+9)*(2101*n^4+33616*n^3+201391*n^2+535416*n+532980)*
%F A025014 a(n+4)+8*(4*n+19)*(5*n+19)*(5*n+14)*(5*n+9)*(2*n+9)*(4*n+17)*(4*n+13)*(4*n+9)*(5*n+18)*(n+5)*a(n+5) = 0. - _Doron Zeilberger_, Apr 02 2013.
%F A025014 a(n) ~ 9^n * sqrt(3/(40*Pi*n)). - _Vaclav Kotesovec_, Aug 09 2013
%F A025014 a(n) = Sum_{k = 0..floor(4*n/9)} (-1)^k * binomial(n, k)*binomial(5*n-9*k-1, n-1). - _Peter Bala_, Oct 16 2024
%p A025014 seq(add((-1)^k * binomial(n, k)*binomial(5*n-9*k-1, n-1), k = 0..floor(4*n/9)), n = 0..20) ; # _Peter Bala_, Oct 16 2024
%t A025014 Flatten[{1,Table[Coefficient[Expand[Sum[x^j,{j,0,8}]^n],x^(4*n)],{n,1,20}]}] (* _Vaclav Kotesovec_, Aug 09 2013 *)
%Y A025014 Column 4 of A201552. Row 9 of A077042.
%K A025014 easy,nonn
%O A025014 0,3
%A A025014 _David W. Wilson_