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A118787 Triangle where T(n,k) = n!*[x^k] ( x/(2*x + log(1-x)) )^(n+1), for n>=k>=0, read by rows. 0
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 (list; table; graph; refs; listen; history; text; internal format)
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 * A191783 A098930 A075372

Adjacent sequences:  A118784 A118785 A118786 * A118788 A118789 A118790

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Apr 29 2006

STATUS

approved

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Last modified December 7 22:50 EST 2021. Contains 349590 sequences. (Running on oeis4.)