A018901 Central hexanomial coefficients: largest coefficient of (1 + x + ... + x^5)^n.

1, 1, 6, 27, 146, 780, 4332, 24017, 135954, 767394, 4395456, 25090131, 144840476, 833196442, 4836766584, 27981391815, 163112472594, 947712321234, 5542414273884, 32312202610863, 189456975899496, 1107575676600876

Offset: 0

Author

Jonn Dalton jdalton(AT)vnet.ibm.com

Keywords

• nonn

Comments

Greatest multiplicity of one- or two-dimensional standard representation of Lie algebras sl(2) in decomposition of tensor power F6^k, where F6 is the standard 6-dimensional irreducible representation of sl(2). - Leonid Bedratyuk, Jul 22 2004

Sum_{k=0..floor(5*n/12)} (-1)^k*binomial(n,k)*binomial(n + floor(5*n/2) - 6*k - 1, n-1). - Warut Roonguthai, May 21 2006

Examples

			Number of ways of getting most likely sum using n 6-sided dice (e.g., for n=2, 7 is the most prevalent sum and there are 6 different ways to get it: 1+6, 2+5, 3+4, 4+3, 5+2, 6+1).
		

Links

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

Crossrefs

Cf. A063419 (bisection). Row 6 of A077042.

Programs

Maple

sum((-1)^(k)*binomial(n,k)*binomial(n+floor(5*n/2)-6*k-1, n-1), k=0..floor(5*n/12)); # Warut Roonguthai, May 21 2006

Mathematica

Flatten[{1,Table[Coefficient[Expand[Sum[x^j,{j,0,5}]^n],x^Floor[5*n/2]],{n,1,20}]}] (* Vaclav Kotesovec, Aug 09 2013 *)

Formula

a(n) ~ 6^n * sqrt(6/(35*Pi*n)). - Vaclav Kotesovec, Aug 09 2013