0,5

Row sums are A118789, where Sum_{n>=0} A118789(n)*x^n/n! = exp( Sum_{n>=1} A032188(n)*x^n/n! ). Main diagonal is A032188(n) = number of labeled series-reduced mobiles (circular rooted trees) with n leaves. Secondary diagonal is A118790.

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

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

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

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

Triangle begins:

1;

1, 1;

1, 3, 5;

1, 6, 23, 41;

1, 10, 65, 255, 469;

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

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

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

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

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

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

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

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

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

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

(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)}

Cf. A118789, A118790, A032188.

Sequence in context: A061649 A073365 A065077 this_sequence A028510 A122053 A124084

Adjacent sequences: A118785 A118786 A118787 this_sequence A118789 A118790 A118791

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Apr 29 2006