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.

1, 2, 1, 8, 12, 1, 32, 120, 30, 1, 128, 896, 560, 56, 1, 512, 5760, 6720, 1680, 90, 1, 2048, 33792, 63360, 29568, 3960, 132, 1, 8192, 186368, 512512, 384384, 96096, 8008, 182, 1, 32768, 983040, 3727360, 4100096, 1647360, 256256, 14560, 240, 1

Offset: 0

Author

Geoffrey Critzer, Jun 07 2020

Keywords

• nonn
• tabl

Examples

			1,
2,   1,
8,   12,   1,
32,  120,  30,   1,
128, 896,  560,  56,   1,
512, 5760, 6720, 1680, 90, 1
		

Crossrefs

Cf. A054879 (row sums), A081294 (column k=0).

Programs

Mathematica

nn = 8; ev[x_] := Sum[x^n/((2 n)!/2^n), {n, 0, nn}];
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

Formula

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.