A364708 Triangle of coefficient of the series reversion in t of the power series (exp(log(1+t*x)/x)-1)*exp(-t) as an e.g.f.

1, 1, 1, 2, 6, 1, 6, 35, 22, 1, 24, 225, 310, 65, 1, 120, 1624, 3885, 1975, 171, 1, 720, 13132, 47929, 45080, 10367, 420, 1, 5040, 118124, 606060, 909489, 409416, 48034, 988, 1, 40320, 1172700, 7995455, 17445645, 13033398, 3152520, 204423, 2259, 1, 362880, 12753576, 110917400, 330281930, 369520305, 153751773, 21587950, 819120, 5065, 1

Offset: 1

Author

Paul Laubie, Oct 20 2023

Keywords

• nonn
• tabl
• easy

Comments

T(n,k) is also the dimension of the operad FMan in arity n with k commutative products.

The sum of each row is n^(n-1).

Examples

			Triangle T(n,k) begins:
n\k  0     1    2   3   4  ...
1    1;
2    1,    1;
3    2,    6,   1;
4    6,   35,  22,  1;
5   24,  225, 310, 65,  1;
...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
• Paul Laubie, Hypertrees and embedding of the FMan operad, arXiv:2401.17439 [math.QA], 2024.

Crossrefs

Column k=0 is A000142(n-1).

Row sums are A000169.

Seems related to a signed version of A079510.

Programs

PARI

T(n) = my(x='x+O('x^(n+1))); [Vecrev(p) | p<-Vec(serlaplace( serreverse((exp(log(1+x*y)/y)-1)*exp(-x) )))]
{my(A=T(10)); for(n=1, #A, print(A[n]))} \\ Andrew Howroyd, Oct 20 2023

Formula

T(n,0) = (n-1)!.

T(n,n-1) = 1.