A362897 - OEIS

%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