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A187914 Generalized Riordan array based on the binomial transform of the Fine's numbers A000957.

%I A187914 #4 Mar 30 2012 18:59:28
%S A187914 1,1,1,2,3,1,6,10,4,1,21,36,15,6,1,79,137,58,29,7,1,311,543,232,132,
%T A187914 37,9,1,1265,2219,954,590,179,57,10,1,5275,9285,4010,2628,837,315,68,
%U A187914 12,1,22431,39587,17156,11732,3861,1629,396,94,13,1,96900,171369,74469,52608,17726,8127,2133,612,108,15,1
%N A187914 Generalized Riordan array based on the binomial transform of the Fine's numbers A000957.
%C A187914 Row sums are A033321(n+1). Second column is A002212(n+1). Equal to A007318*A187913.
%F A187914 Let g(x)=(1+x-sqrt(1-6x+5x^2))/(2x(2-x)) be the g.f. of A033321, the binomial transform of the Fine numbers.
%F A187914 Then the g.f. of the k-th column is x^k*g(x)^((k+2)/2)/(1-2*x*g(x))^(k/2) if k is even, and
%F A187914 x^k*g(x)^((k+1)/2)/(1-2*x*g(x))^((k+1)/2) if k is odd. Otherwise put, column k has g.f.
%F A187914 g.f. x^k*g(x)^(k+1)/(1-xg(x)-x^2g(x)^2)^floor((k+1)/2).
%e A187914 Triangle begins
%e A187914 1,
%e A187914 1, 1,
%e A187914 2, 3, 1,
%e A187914 6, 10, 4, 1,
%e A187914 21, 36, 15, 6, 1,
%e A187914 79, 137, 58, 29, 7, 1,
%e A187914 311, 543, 232, 132, 37, 9, 1,
%e A187914 1265, 2219, 954, 590, 179, 57, 10, 1,
%e A187914 5275, 9285, 4010, 2628, 837, 315, 68, 12, 1,
%e A187914 22431, 39587, 17156, 11732, 3861, 1629, 396, 94, 13, 1
%e A187914 Production matrix is
%e A187914 1, 1,
%e A187914 1, 2, 1,
%e A187914 1, 2, 1, 1,
%e A187914 1, 2, 1, 2, 1,
%e A187914 1, 2, 1, 2, 1, 1,
%e A187914 1, 2, 1, 2, 1, 2, 1,
%e A187914 1, 2, 1, 2, 1, 2, 1, 1,
%e A187914 1, 2, 1, 2, 1, 2, 1, 2, 1,
%e A187914 1, 2, 1, 2, 1, 2, 1, 2, 1, 1;
%e A187914 Hence, for instance, we have
%e A187914 79=1*0+1.21+1.36+1.15+1.6+1.1;
%e A187914 137=1.21+2.36+2.15+2.6+2.1;
%e A187914 58=1.36+1.15+1.6+1.1
%K A187914 nonn,easy,tabl
%O A187914 0,4
%A A187914 _Paul Barry_, Mar 15 2011