A369919
Triangular array read by rows. T(n,k) is the number of labeled posets on [n] of rank at most one with exactly k elements of positive indegree, n >= 0, 0 <= k <= max{0,n-1}.
Original entry on oeis.org
1, 1, 1, 2, 1, 9, 3, 1, 28, 54, 4, 1, 75, 490, 270, 5, 1, 186, 3375, 6860, 1215, 6, 1, 441, 20181, 118125, 84035, 5103, 7, 1, 1016, 111132, 1668296, 3543750, 941192, 20412, 8, 1, 2295, 580644, 21003948, 116363646, 95681250, 9882516, 78732, 9
Offset: 0
Triangle begins
1;
1;
1, 2;
1, 9, 3;
1, 28, 54, 4;
1, 75, 490, 270, 5;
1, 186, 3375, 6860, 1215, 6;
...
-
nn = 9; Map[Select[#, # > 0 &] &,Table[n!, {n, 0, nn}] CoefficientList[Series[ Sum[ Exp[y x]^(2^n - 1) x^n/n!, {n, 0, nn}], {x, 0, nn}], {x, y}]] // Grid
A369921
Number of cover relations summed over the rank-1 labeled posets on [n].
Original entry on oeis.org
0, 0, 2, 18, 204, 2940, 56670, 1471806, 52067512, 2520298584, 167850357210, 15435027907530, 1967345286257604, 348527628228821652, 86057693880611800438, 29677160119074814383030, 14321851348104417100842480
Offset: 0
-
nn = 16; Table[Table[n!, {n, 0, nn}] CoefficientList[D[Series[Sum[Exp[y x]^Binomial[n, i]*Exp[ x]^(2^n - Binomial[n, i] - 1) x^n/n!, {n, 0, nn}], {x, 0, nn}], y] /. y -> 1, x]*i, {i, 1, nn - 1}] // Total
A370208
Triangular array read by rows. T(n,k) is the number of idempotent binary relations on [n] having no proper power primitive (A360718) with exactly k irreflexive points.
Original entry on oeis.org
1, 1, 1, 3, 6, 13, 39, 87, 348, 24, 841, 4205, 480, 11643, 69858, 9420, 240, 227893, 1595251, 206640, 9240, 6285807, 50286456, 5389552, 299040, 3360, 243593041, 2192337369, 172041408, 9848160, 211680
Offset: 0
Triangle begins
1;
1, 1;
3, 6;
13, 39;
87, 348, 24;
841, 4205, 480;
11643, 69858, 9420, 240;
227893, 1595251, 206640, 9240;
...
-
nn = 9; A[x_] := Sum[x^n/n! Exp[(2^n - 1) x], {n, 0, nn}];
c[x_] := Log[A[x]] - x; Map[Select[#, # > 0 &] &,
Range[0, nn]! CoefficientList[
Series[2 (Exp[ y x D[c[ x], x]/2] - 1) Exp[c[x]] Exp[ x] +
Exp[c[ x]] (y x Exp[ x] + Exp[ x]), {x, 0, nn}], {x, y}]]
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