A025013 Central octonomial coefficients: largest coefficient of (1+x+...+x^7)^n.

1, 1, 8, 48, 344, 2460, 18152, 134512, 1012664, 7635987, 58199208, 443658688, 3409213016, 26184550496, 202384723528, 1562970918720, 12133130451576, 94094281551304, 732910480638272, 5702603044247504, 44538031693977544

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

• Vaclav Kotesovec, Table of n, a(n) for n = 0..500 (first 200 terms from T. D. Noe)
• Vaclav Kotesovec, Recurrence.
• Index entries for sequences of k-nomial coefficients

Crossrefs

Row 8 of A077042.

Programs

Maple

seq(add((-1)^k * binomial(n, k)*binomial(n+floor(7*n/2)-8*k-1, n-1), k = 0..floor(n/2) ), n = 0..20); # Peter Bala, Oct 15 2024

Mathematica

Flatten[{1,Table[Coefficient[Expand[Sum[x^j,{j,0,7}]^n],x^Floor[7*n/2]],{n,1,20}]}] (* Vaclav Kotesovec, Aug 09 2013 *)

Formula

a(n) ~ 8^n * sqrt(2/(21*Pi*n)). - Vaclav Kotesovec, Aug 09 2013

a(n) = Sum_{k = 0..floor(n/2)} (-1)^k * binomial(n, k)*binomial(n+floor(7*n/2)-8*k-1, n-1). - Peter Bala, Oct 15 2024