A118788 - OEIS

%I #7 Apr 30 2014 14:24:38

%S 1,1,1,1,3,5,1,6,23,41,1,10,65,255,469,1,15,145,930,3679,6889,1,21,

%T 280,2590,16429,65247,123605,1,28,490,6090,54789,344694,1371887,

%U 2620169,1,36,798,12726,151599,1338330,8367785,33347535,64074901,1,45,1230,24360

%N 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 Row sums are A118789, where Sum_{n>=0} A118789(n)*x^n/n! = exp( Sum_{n>=1} A032188(n)*x^n/n! ).

%C Main diagonal is A032188(n) = number of labeled series-reduced mobiles (circular rooted trees) with n leaves.

%C Secondary diagonal is A118790.

%F Main diagonal has e.g.f.: series_reversion[2*x+log(1-x)].

%e Row sums e.g.f. equals the exponential of the diagonal e.g.f.:

%e 1 + x + 2*x^2/2! + 9*x^3/3! + 71*x^4/4! +...+ A118789(n)*x^n/n! +...

%e = exp(x + x^2/2! + 5*x^3/3! + 41*x^4/4! +...+ A032188(n)*x^n/n! +...).

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 3, 5;

%e 1, 6, 23, 41;

%e 1, 10, 65, 255, 469;

%e 1, 15, 145, 930, 3679, 6889;

%e 1, 21, 280, 2590, 16429, 65247, 123605;

%e 1, 28, 490, 6090, 54789, 344694, 1371887, 2620169;

%e 1, 36, 798, 12726, 151599, 1338330, 8367785, 33347535, 64074901; ...

%e Triangle is formed from powers of F(x) = x/(2*x + log(1-x)):

%e F(x)^1 = (1) + 1/2*x + 7/12*x^2 + 17/24*x^3 + 629/720*x^4 +...

%e F(x)^2 = (1 + x) + 17/12*x^2 + 2*x^3 + 671/240*x^4 +...

%e F(x)^3 = (1 + 3/2*x + 5/2*x^2) + 4*x^3 + 1489/240*x^4 +...

%e F(x)^4 = (1 + 6/3*x + 23/6*x^2 + 41/6*x^3) + 8351/720*x^4 +...

%e F(x)^5 = (1 + 10/4*x + 65/12*x^2 + 255/24*x^3 + 469/24*x^4) +...

%o (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 Cf. A118789, A118790, A032188.

%Y Third column is A241765.

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Apr 29 2006