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.

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

Table of n, a(n) for n=0..30.

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

Adjacent sequences: A370205 A370206 A370207 * A370209 A370210 A370211

Keyword

nonn,tabf

Author

Geoffrey Critzer, Feb 11 2024

Status

approved