%I #7 May 10 2023 22:39:21
%S 1,1,1,1,1,1,1,1,3,1,1,1,7,7,1,1,1,13,74,19,1,1,1,22,638,1474,47,1,1,
%T 1,34,4663,118949,41876,130,1,1,1,50,28529,7643021,42483668,1540696,
%U 343,1,1,1,70,151600,396979499,33179970333,23524514635,68343112,951,1
%N Array read by antidiagonals: T(n,k) is the number of nonisomorphic multisets of endofunctions on an n-set with k endofunctions.
%C Isomorphism is up to permutations of the elements of the n-set.
%H Andrew Howroyd, <a href="/A362897/b362897.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals).
%F T(0,k) = T(1,k) = 1.
%e Array begins:
%e ======================================================================
%e n/k| 0 1 2 3 4 5 ...
%e ---+------------------------------------------------------------------
%e 0 | 1 1 1 1 1 1 ...
%e 1 | 1 1 1 1 1 1 ...
%e 2 | 1 3 7 13 22 34 ...
%e 3 | 1 7 74 638 4663 28529 ...
%e 4 | 1 19 1474 118949 7643021 396979499 ...
%e 5 | 1 47 41876 42483668 33179970333 20762461502595 ...
%e 6 | 1 130 1540696 23524514635 274252613077267 2559276179593762172 ...
%e ...
%o (PARI)
%o permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
%o K(v,m) = {prod(i=1, #v, my(g=gcd(v[i],m), e=v[i]/g); sum(j=1, #v, my(t=v[j]); if(e%(t/gcd(t,m))==0, t))^g)}
%o T(n,k) = {if(n==0, 1, my(s=0); forpart(q=n, s+=permcount(q) * polcoef(exp(sum(m=1, k, K(q,m)*x^m/m, O(x*x^k))), k)); s/n!)}
%Y Columns k=0..3 are A000012, A001372, A054745, A362898.
%Y Row n=2 is A002623.
%Y Main diagonal is A277839.
%Y Cf. A362644.
%K nonn,tabl
%O 0,9
%A _Andrew Howroyd_, May 10 2023