A118787 Triangle where T(n,k) = n!*[x^k] ( x/(2*x + log(1-x)) )^(n+1), for n>=k>=0, read by rows.

1, 1, 1, 2, 3, 5, 6, 12, 23, 41, 24, 60, 130, 255, 469, 120, 360, 870, 1860, 3679, 6889, 720, 2520, 6720, 15540, 32858, 65247, 123605, 5040, 20160, 58800, 146160, 328734, 689388, 1371887, 2620169, 40320, 181440, 574560, 1527120, 3638376, 8029980

Offset: 0

Author

Paul D. Hanna, Apr 29 2006

Keywords

• nonn
• tabl

Comments

Row sums are A112487. Main diagonal is A032188(n) = number of labeled series-reduced mobiles (circular rooted trees) with n leaves.

Examples

			Triangle begins:
1;
1, 1;
2, 3, 5;
6, 12, 23, 41;
24, 60, 130, 255, 469;
120, 360, 870, 1860, 3679, 6889;
720, 2520, 6720, 15540, 32858, 65247, 123605;
5040, 20160, 58800, 146160, 328734, 689388, 1371887, 2620169; ...
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)/1! +17/12*x^2 + 2*x^3 + 671/240*x^4 ...
F(x)^3 = (2 + 3*x + 5*x^2)/2! + 4*x^3 + 1489/240*x^4 +...
F(x)^4 = (6 + 12*x + 23*x^2 + 41/6*x^3)/3! + 8351/720*x^4 +...
F(x)^5 = (24 + 60*x + 130*x^2 + 255*x^3 + 469*x^4)/4! +...
		

Crossrefs

Cf. A118788, A112487, A032188.

Programs

PARI

{T(n,k)=local(x=X+X^2*O(X^(k+2)));n!*polcoeff((x/(2*x+log(1-x)))^(n+1),k,X)}

Formula

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