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%I #62 Jul 01 2020 23:33:36
%S 1,2,1,8,12,1,32,120,30,1,128,896,560,56,1,512,5760,6720,1680,90,1,
%T 2048,33792,63360,29568,3960,132,1,8192,186368,512512,384384,96096,
%U 8008,182,1,32768,983040,3727360,4100096,1647360,256256,14560,240,1
%N Triangular array read by rows. Let P be the poset of all even sized subsets of [2n] ordered by inclusion. T(n,k) is the number of intervals in P with length k, 0<=k<=n, n>=0.
%F Let E(x) = Sum_{n>=0} x^n/((2n)!/2^n). Then Sum_{n>=0} Sum{k=0..n} T(n,k) y^k*x^n/((2n)!/2^n) = E(y*x) * E(x)^2.
%e 1,
%e 2, 1,
%e 8, 12, 1,
%e 32, 120, 30, 1,
%e 128, 896, 560, 56, 1,
%e 512, 5760, 6720, 1680, 90, 1
%t nn = 8; ev[x_] := Sum[x^n/((2 n)!/2^n), {n, 0, nn}];
%t Map[Select[#, # > 0 &] &, Table[(2 n)!/2^n, {n, 0, nn}] CoefficientList[Series[ev[x]^2 ev[y x], {x, 0, nn}], {x, y}]] // Flatten
%Y Cf. A054879 (row sums), A081294 (column k=0).
%K nonn,tabl
%O 0,2
%A _Geoffrey Critzer_, Jun 07 2020