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%I #17 Feb 19 2022 14:17:00
%S 1,2,7,2,37,21,6,261,232,108,24,2301,2935,1760,660,120,24343,42396,
%T 30630,14880,4680,720,300455,692055,586572,335790,139440,37800,5040,
%U 4238153,12631200,12387592,8008896,3959760,1438080,342720,40320
%N 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 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 Array begins
%e 1;
%e 2;
%e 7, 2;
%e 37, 21, 6;
%e 261, 232, 108, 24;
%e 2301, 2935, 1760, 660, 120;
%e 24343, 42396, 30630, 14880, 4680, 720;
%e ...
%t 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 Cf. A072597 (column k=1), A000169(n+1) (row sums).
%K nonn,tabf
%O 0,2
%A _Geoffrey Critzer_, Feb 13 2022
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