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

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, 1, 34, 4663, 118949, 41876, 130, 1, 1, 1, 50, 28529, 7643021, 42483668, 1540696, 343, 1, 1, 1, 70, 151600, 396979499, 33179970333, 23524514635, 68343112, 951, 1

Offset

0,9

Comments

Isomorphism is up to permutations of the elements of the n-set.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals).

Formula

T(0,k) = T(1,k) = 1.

Example

Array begins:
======================================================================
n/k| 0   1       2           3               4                   5 ...
---+------------------------------------------------------------------
0  | 1   1       1           1               1                   1 ...
1  | 1   1       1           1               1                   1 ...
2  | 1   3       7          13              22                  34 ...
3  | 1   7      74         638            4663               28529 ...
4  | 1  19    1474      118949         7643021           396979499 ...
5  | 1  47   41876    42483668     33179970333      20762461502595 ...
6  | 1 130 1540696 23524514635 274252613077267 2559276179593762172 ...
  ...

Prog

(PARI)
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}
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)}
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!)}

Crossrefs

Columns k=0..3 are A000012, A001372, A054745, A362898.

Row n=2 is A002623.

Main diagonal is A277839.

Cf. A362644.

Sequence in context: A081297 A110180 A328718 * A005765 A360289 A343717

Adjacent sequences: A362894 A362895 A362896 * A362898 A362899 A362900

Keyword

nonn,tabl

Author

Andrew Howroyd, May 10 2023

Status

approved