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A362897 Array read by antidiagonals: T(n,k) is the number of nonisomorphic multisets of endofunctions on an n-set with k endofunctions.

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