login
Number of two-rowed partitions of length 4.
3

%I #14 Apr 05 2016 01:00:25

%S 1,1,3,5,10,15,26,38,60,85,125,172,243,325,442,580,767,986,1275,1612,

%T 2045,2548,3179,3910,4812,5849,7109,8554,10285,12259,14599,17255,

%U 20372,23895,27991,32603,37925,43890,50725,58361,67053,76727,87678,99825,113503

%N Number of two-rowed partitions of length 4.

%H G. E. Andrews, <a href="http://dx.doi.org/10.1007/PL00001284">MacMahon's Partition Analysis II: Fundamental Theorems</a>, Annals Combinatorics, 4 (2000), 327-338.

%H L. Colmenarejo, <a href="http://arxiv.org/abs/1604.00803">Combinatorics on several families of Kronecker coefficients related to plane partitions</a>, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 5.

%F G.f.: 1/((1-x)*((1-x^2)*...*(1-x^m))^2*(1-x^(m+1))) for m = 4.

%p a:= n-> (Matrix(24, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 2, 0, -1, -4, -2, 1, 5, 6, 0, -4, -6, -4, 0, 6, 5, 1, -2, -4, -1, 0, 2, 1, -1][i] else 0 fi)^n)[1,1]: seq (a(n), n=0..50); # _Alois P. Heinz_, Jul 31 2008

%t m = 4; n = 45; gf = 1/((1-x)*Product[1-x^k, {k, 2, m}]^2*(1-x^(m+1))) + O[x]^n; CoefficientList[gf, x] (* _Jean-François Alcover_, Jul 17 2015 *)

%Y Cf. A008763, A001993, A070558, A070559.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, May 07 2002