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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.
0
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
0,4
LINKS
David Rosenblatt, On the graphs of finite Boolean relation matrices, Journal of Research, National Bureau of Standards, Vol 67B No. 4 Oct-Dec 1963.
FORMULA
E.g.f.: 2(exp(y*x*c'(x)/2)-1)*exp(c(x))*exp(x) + exp(c(x))*(y*x*exp(x) + exp(x)) where c(x) is the e.g.f. for A002031.
EXAMPLE
Triangle begins
1;
1, 1;
3, 6;
13, 39;
87, 348, 24;
841, 4205, 480;
11643, 69858, 9420, 240;
227893, 1595251, 206640, 9240;
...
MATHEMATICA
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}]]
CROSSREFS
Cf. A360718 (row sums), A001831 (column k=0), A360743 (T(n,0) + T(n,1) ), A151817 (T(2n,n) for n>=2), A002031.
Sequence in context: A264236 A216999 A036781 * A084816 A055738 A301656
KEYWORD
nonn,tabf
AUTHOR
Geoffrey Critzer, Feb 11 2024
STATUS
approved