cp's OEIS Frontend

A245767 Triangular array read by rows: T(n,k) is the number of transitive relations on {1,2,...,n} that have exactly k reflexive points, n>=0, 0<=k<=n.

%I A245767 #19 Feb 25 2022 08:35:38
%S A245767 1,1,1,3,6,4,19,57,66,29,219,876,1428,1116,355,4231,21155,44500,49070,
%T A245767 28405,6942,130023,780138,2013810,2858700,2354415,1068576,209527,
%U A245767 6129859,42909013,131457522,228345565,242894155,158322528,58628647,9535241
%N A245767 Triangular array read by rows: T(n,k) is the number of transitive relations on {1,2,...,n} that have exactly k reflexive points, n>=0, 0<=k<=n.
%C A245767 Row sums give A006905.
%C A245767 Column k=0 is A001035.
%C A245767 T(n,n) = A000798(n).
%H A245767 Alois P. Heinz, <a href="/A245767/b245767.txt">Rows n = 0..18, flattened</a>
%F A245767 E.g.f.: A(x + exp(y*x) - 1) where A(x) is the e.g.f. for A001035.
%e A245767 T(2,1) = 6 because we have: {(1,1)}, {(2,2)}, {(1,1),(1,2)}, {(1,1),(2,1)}, {(2,2),(1,2)}, {(2,2),(2,1)}.
%e A245767 Triangle T(n,k) begins:
%e A245767        1;
%e A245767        1,      1;
%e A245767        3,      6,       4;
%e A245767       19,     57,      66,      29;
%e A245767      219,    876,    1428,    1116,     355;
%e A245767     4231,  21155,   44500,   49070,   28405,    6942;
%e A245767   130023, 780138, 2013810, 2858700, 2354415, 1068576, 209527;
%e A245767   ...
%t A245767 A001035 = Cases[Import["https://oeis.org/A001035/b001035.txt", "Table"], {_, _}][[All, 2]];
%t A245767 lg = Length[A001035];
%t A245767 A[x_] = Sum[A001035[[n+1]] x^n/n!, {n, 0, lg-1}];
%t A245767 CoefficientList[#, y]& /@ (CoefficientList[A[x + Exp[y*x]-1] + O[x]^lg, x]* Range[0, lg-1]!) // Flatten (* _Jean-François Alcover_, Jan 01 2020 *)
%Y A245767 Cf. A000798, A001035, A006905.
%K A245767 nonn,tabl
%O A245767 0,4
%A A245767 _Geoffrey Critzer_, Jul 31 2014