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Central hexanomial coefficients: largest coefficient of (1 + x + ... + x^5)^n.
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%I #27 Feb 18 2019 08:52:44

%S 1,1,6,27,146,780,4332,24017,135954,767394,4395456,25090131,144840476,

%T 833196442,4836766584,27981391815,163112472594,947712321234,

%U 5542414273884,32312202610863,189456975899496,1107575676600876

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

%C 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

%C 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

%H T. D. Noe, <a href="/A018901/b018901.txt">Table of n, a(n) for n = 0..200</a>

%H Vaclav Kotesovec, <a href="/A018901/a018901.txt">Recurrence</a>

%F a(n) ~ 6^n * sqrt(6/(35*Pi*n)). - _Vaclav Kotesovec_, Aug 09 2013

%e 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).

%p 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

%t 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 *)

%Y Cf. A001405, A002426, A005190, A005191, A025012, A025013, A025014.

%K nonn

%O 0,3

%A Jonn Dalton jdalton(AT)vnet.ibm.com