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

0,4

Comments

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

Links

Table of n, a(n) for n=0..41.

Formula

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

Example

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! +...

Prog

(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)}

Crossrefs

Cf. A118788, A112487, A032188.

Sequence in context: A128958 A007435 A125877 * A359668 A191783 A358290

Adjacent sequences: A118784 A118785 A118786 * A118788 A118789 A118790

Keyword

nonn,tabl

Author

Paul D. Hanna, Apr 29 2006

Status

approved