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

0,3

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

References

Rudolph-Lilith, Michelle, and Lyle E. Muller. "On a link between Dirichlet kernels and central multinomial coefficients." Discrete Mathematics 338.9 (2015): 1567-1572.

Links

T. D. Noe, Table of n, a(n) for n=0..200

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

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

Cf. A001405, A002426, A005190, A005191, A018901, A025012, A025013

Sequence in context: A162769 A126504 A361280 * A246567 A322086 A075139

Adjacent sequences: A025011 A025012 A025013 * A025015 A025016 A025017

Keyword

easy,nonn

Author

David W. Wilson

Status

approved