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A328821 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.

%I A328821 #62 Jul 01 2020 23:33:36
%S A328821 1,2,1,8,12,1,32,120,30,1,128,896,560,56,1,512,5760,6720,1680,90,1,
%T A328821 2048,33792,63360,29568,3960,132,1,8192,186368,512512,384384,96096,
%U A328821 8008,182,1,32768,983040,3727360,4100096,1647360,256256,14560,240,1
%N A328821 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 A328821 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 A328821 1,
%e A328821 2,   1,
%e A328821 8,   12,   1,
%e A328821 32,  120,  30,   1,
%e A328821 128, 896,  560,  56,   1,
%e A328821 512, 5760, 6720, 1680, 90, 1
%t A328821 nn = 8; ev[x_] := Sum[x^n/((2 n)!/2^n), {n, 0, nn}];
%t A328821 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 A328821 Cf. A054879 (row sums), A081294 (column k=0).
%K A328821 nonn,tabl
%O A328821 0,2
%A A328821 _Geoffrey Critzer_, Jun 07 2020