OFFSET
0,3
COMMENTS
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
LINKS
T. D. Noe, Table of n, a(n) for n=0..200
Michelle Rudolph-Lilith and Lyle E. Muller, On a link between Dirichlet kernels and central multinomial coefficients, Discrete Mathematics 338.9 (2015): 1567-1572.
FORMULA
The Almkvist-Zeilberger algorithm in EKHAD establishes the following recurrence:
-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)*(
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
*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)*
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.
a(n) ~ 9^n * sqrt(3/(40*Pi*n)). - Vaclav Kotesovec, Aug 09 2013
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
MAPLE
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
MATHEMATICA
Flatten[{1, Table[Coefficient[Expand[Sum[x^j, {j, 0, 8}]^n], x^(4*n)], {n, 1, 20}]}] (* Vaclav Kotesovec, Aug 09 2013 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
STATUS
approved