A245767 - OEIS

%I #19 Feb 25 2022 08:35:38

%S 1,1,1,3,6,4,19,57,66,29,219,876,1428,1116,355,4231,21155,44500,49070,

%T 28405,6942,130023,780138,2013810,2858700,2354415,1068576,209527,

%U 6129859,42909013,131457522,228345565,242894155,158322528,58628647,9535241

%N 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 Row sums give A006905.

%C Column k=0 is A001035.

%C T(n,n) = A000798(n).

%H Alois P. Heinz, <a href="/A245767/b245767.txt">Rows n = 0..18, flattened</a>

%F E.g.f.: A(x + exp(y*x) - 1) where A(x) is the e.g.f. for A001035.

%e 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 Triangle T(n,k) begins:

%e        1;

%e        1,      1;

%e        3,      6,       4;

%e       19,     57,      66,      29;

%e      219,    876,    1428,    1116,     355;

%e     4231,  21155,   44500,   49070,   28405,    6942;

%e   130023, 780138, 2013810, 2858700, 2354415, 1068576, 209527;

%e   ...

%t A001035 = Cases[Import["https://oeis.org/A001035/b001035.txt", "Table"], {_, _}][[All, 2]];

%t lg = Length[A001035];

%t A[x_] = Sum[A001035[[n+1]] x^n/n!, {n, 0, lg-1}];

%t 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 Cf. A000798, A001035, A006905.

%K nonn,tabl

%O 0,4

%A _Geoffrey Critzer_, Jul 31 2014