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A341093 Triangular array read by rows. T(n,k) is the number of partial functions on [n] with index k, n=0 implies k=1, otherwise n >= 1, 1 <= k <= n.

%I A341093 #17 Feb 19 2022 14:17:00
%S A341093 1,2,7,2,37,21,6,261,232,108,24,2301,2935,1760,660,120,24343,42396,
%T A341093 30630,14880,4680,720,300455,692055,586572,335790,139440,37800,5040,
%U A341093 4238153,12631200,12387592,8008896,3959760,1438080,342720,40320
%N A341093 Triangular array read by rows. T(n,k) is the number of partial functions on [n] with index k, n=0 implies k=1, otherwise n >= 1, 1 <= k <= n.
%C A341093 For every partial function f, there are smallest positive integers k,m such that f^k = f^(k+m). The integer k is the index of f.
%e A341093 Array begins
%e A341093       1;
%e A341093       2;
%e A341093       7,     2;
%e A341093      37,    21,     6;
%e A341093     261,   232,   108,    24;
%e A341093    2301,  2935,  1760,   660,  120;
%e A341093   24343, 42396, 30630, 14880, 4680, 720;
%e A341093   ...
%t A341093 nn = 8; np = Exp[NestList[x Exp[#] &, x, nn]]; fp = Exp[Log[1/(1 - NestList[x Exp[#] &, x Exp[x], nn])]];Map[Select[#, # > 0 &] &,Prepend[Table[Range[0, nn]! CoefficientList[Series[(fp[[k + 1]] - fp[[k]])*(np[[k + 1]]) + (fp[[k + 1]])*(np[[k + 1]] - np[[k]]) - (fp[[k + 1]] - fp[[k]]) (np[[k + 1]] - np[[k]]), {x, 0, nn}], x], {k, 1, nn - 1}], Range[0, nn]! CoefficientList[Series[1/(1 - x Exp[x])*Exp[x], {x, 0, nn}], x]] // Transpose] // Grid
%Y A341093 Cf. A072597 (column k=1), A000169(n+1) (row sums).
%K A341093 nonn,tabf
%O A341093 0,2
%A A341093 _Geoffrey Critzer_, Feb 13 2022