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A118788 Triangle where T(n,k) = n!/(n-k)!*[x^k] ( x/(2*x + log(1-x)) )^(n+1), for n>=k>=0, read by rows.

%I A118788 #14 Mar 04 2025 22:43:02
%S A118788 1,1,1,1,3,5,1,6,23,41,1,10,65,255,469,1,15,145,930,3679,6889,1,21,
%T A118788 280,2590,16429,65247,123605,1,28,490,6090,54789,344694,1371887,
%U A118788 2620169,1,36,798,12726,151599,1338330,8367785,33347535,64074901,1,45,1230,24360
%N A118788 Triangle where T(n,k) = n!/(n-k)!*[x^k] ( x/(2*x + log(1-x)) )^(n+1), for n>=k>=0, read by rows.
%C A118788 Row sums are A118789, where Sum_{n>=0} A118789(n)*x^n/n! = exp( Sum_{n>=1} A032188(n)*x^n/n! ).
%C A118788 Main diagonal is A032188(n) = number of labeled series-reduced mobiles (circular rooted trees) with n leaves.
%C A118788 Secondary diagonal is A118790.
%F A118788 Main diagonal has e.g.f.: series_reversion[2*x+log(1-x)].
%F A118788 Conjecture: T(n,k) = Sum_{j=0..k} binomial(n+j, n-k)*A269940(k, j) for 0 <= k <= n. - _Mikhail Kurkov_, Feb 17 2025
%e A118788 Row sums e.g.f. equals the exponential of the diagonal e.g.f.:
%e A118788 1 + x + 2*x^2/2! + 9*x^3/3! + 71*x^4/4! +...+ A118789(n)*x^n/n! +...
%e A118788 = exp(x + x^2/2! + 5*x^3/3! + 41*x^4/4! +...+ A032188(n)*x^n/n! +...).
%e A118788 Triangle begins:
%e A118788   1;
%e A118788   1, 1;
%e A118788   1, 3, 5;
%e A118788   1, 6, 23, 41;
%e A118788   1, 10, 65, 255, 469;
%e A118788   1, 15, 145, 930, 3679, 6889;
%e A118788   1, 21, 280, 2590, 16429, 65247, 123605;
%e A118788   1, 28, 490, 6090, 54789, 344694, 1371887, 2620169;
%e A118788   1, 36, 798, 12726, 151599, 1338330, 8367785, 33347535, 64074901;
%e A118788   ...
%e A118788 Triangle is formed from powers of F(x) = x/(2*x + log(1-x)):
%e A118788   F(x)^1 = (1) + 1/2*x + 7/12*x^2 + 17/24*x^3 + 629/720*x^4 +...
%e A118788   F(x)^2 = (1 + x) + 17/12*x^2 + 2*x^3 + 671/240*x^4 +...
%e A118788   F(x)^3 = (1 + 3/2*x + 5/2*x^2) + 4*x^3 + 1489/240*x^4 +...
%e A118788   F(x)^4 = (1 + 6/3*x + 23/6*x^2 + 41/6*x^3) + 8351/720*x^4 +...
%e A118788   F(x)^5 = (1 + 10/4*x + 65/12*x^2 + 255/24*x^3 + 469/24*x^4) +...
%o A118788 (PARI) {T(n,k)=local(x=X+X^2*O(X^(k+2)));n!/(n-k)!*polcoeff((x/(2*x+log(1-x)))^(n+1),k,X)}
%Y A118788 Cf. A118789, A118790, A032188.
%Y A118788 Third column is A241765.
%K A118788 nonn,tabl
%O A118788 0,5
%A A118788 _Paul D. Hanna_, Apr 29 2006