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%I #6 Sep 22 2013 15:58:09
%S 1,0,5,1,36,15,228,231,1313,1939,6971,11899,33118,59543,140620,254476,
%T 538042,959028,1871808,3258512,5981444,10140360,17726166,29257848,
%U 49127549,79032258,128267727,201437596
%N Coefficient of s(2n) in s(n,n) * s(n,n) * s(n,n) * s(n,n) * s(n,n), where s(2n) is the Schur function corresponding to the trivial representation, s(n,n) is a Schur function corresponding two the two row partition and * represents the inner or Kronecker product of symmetric functions.
%D I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Mathematical Monographs, Oxford Univ. Press, second edition, 1995.
%F a(n) = Sum_{gamma} Chi^{(n, n)}( gamma )^5/z(gamma) the sum is over all partitions gamma of 2n Chi^lambda(gamma) is the value of the symmetric group character z(gamma) is the size of the stablizer of the conjugacy class of symmetric group indexed by the partition gamma
%p compsclr := proc(k) local gamma; add( combinat[Chi]( [k,k], gamma)^5/ZEE(gamma),gamma= combinat[partition](2*k)); end: ZEE := proc (mu) local res, m, i; m := 1; res := convert(mu,`*`); for i from 2 to nops(mu) do if mu[i] <> mu[i-1] then m := 1 else m := m+1 fi; res := res*m; od; res; end:
%Y Cf. A008763 for Chi( [k, k], gamma)^4/ZEE(gamma) instead of Chi( [k, k], gamma)^5/ZEE(gamma) in the programs above.
%K nonn
%O 0,3
%A _Mike Zabrocki_, Apr 25 2003
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