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

1, 1, 9, 61, 489, 3951, 32661, 273127, 2306025, 19610233, 167729959, 1441383219, 12434998005, 107632809909, 934263293679, 8129320828911, 70886845397481, 619288973447049, 5419332253680705, 47494787636620701, 416800775902696839

Offset: 0

Author

David W. Wilson

Keywords

• easy
• nonn

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.
• Index entries for sequences of k-nomial coefficients

Crossrefs

Column 4 of A201552. Row 9 of A077042.

Programs

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 *)

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