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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.

%I A364708 #38 Feb 03 2024 10:13:37
%S A364708 1,1,1,2,6,1,6,35,22,1,24,225,310,65,1,120,1624,3885,1975,171,1,720,
%T A364708 13132,47929,45080,10367,420,1,5040,118124,606060,909489,409416,48034,
%U A364708 988,1,40320,1172700,7995455,17445645,13033398,3152520,204423,2259,1,362880,12753576,110917400,330281930,369520305,153751773,21587950,819120,5065,1
%N 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.
%C A364708 T(n,k) is also the dimension of the operad FMan in arity n with k commutative products.
%C A364708 The sum of each row is n^(n-1).
%H A364708 Andrew Howroyd, <a href="/A364708/b364708.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%H A364708 Paul Laubie, <a href="https://arxiv.org/abs/2401.17439">Hypertrees and embedding of the FMan operad</a>, arXiv:2401.17439 [math.QA], 2024.
%F A364708 T(n,0) = (n-1)!.
%F A364708 T(n,n-1) = 1.
%e A364708 Triangle T(n,k) begins:
%e A364708 n\k  0     1    2   3   4  ...
%e A364708 1    1;
%e A364708 2    1,    1;
%e A364708 3    2,    6,   1;
%e A364708 4    6,   35,  22,  1;
%e A364708 5   24,  225, 310, 65,  1;
%e A364708 ...
%o A364708 (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) )))]
%o A364708 {my(A=T(10)); for(n=1, #A, print(A[n]))} \\ _Andrew Howroyd_, Oct 20 2023
%Y A364708 Column k=0 is A000142(n-1).
%Y A364708 Row sums are A000169.
%Y A364708 Seems related to a signed version of A079510.
%K A364708 nonn,tabl,easy
%O A364708 1,4
%A A364708 _Paul Laubie_, Oct 20 2023