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%I #10 Jan 11 2016 20:59:55
%S 1,4,3,16,24,9,64,144,108,27,256,768,864,432,81,1024,3840,5760,4320,
%T 1620,243,4096,18432,34560,34560,19440,5832,729,16384,86016,193536,
%U 241920,181440,81648,20412,2187,65536,393216,1032192,1548288
%N Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j)*3^j.
%C T(i,j) is the number of sequences (X_1, X_2, X_3) of subsets of {1,2,...,i} such that X_1 intersect X_2 intersect X_3 is empty and X_3 contains exactly j elements. Cf. Stanley reference. - _Geoffrey Critzer_, Jan 11 2016
%D B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.
%D R. P. Stanley, Enumerative Combinatorics Vol I, Cambridge Univ. Press, 1997, page 11.
%F E.g.f.: exp(4*x + 3*y*x). - _Geoffrey Critzer_, Jan 11 2016
%e 1;
%e 4, 3;
%e 16, 24, 9;
%e 64, 144, 108, 27;
%e 256, 768, 864, 432, 81;
%e 1024, 3840, 5760, 4320, 1620, 243;
%e 4096, 18432, 34560, 34560, 19440, 5832, 729;
%t nn = 10; Map[Select[#, # > 0 &] &,Range[0, nn]! CoefficientList[
%t Series[Exp[3 x + 3 y x] Exp[x], {x, 0, nn}], {x, y}]] // Grid (* _Geoffrey Critzer_, Jan 11 2016 *)
%Y Cf. A038207, A038285.
%K nonn,tabl,easy
%O 0,2
%A _N. J. A. Sloane_.
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