A118788 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.

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, 1, 45, 1230, 24360

Offset

0,5

Comments

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.

Links

Table of n, a(n) for n=0..48.

Formula

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

Example

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) +...

Prog

(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)}

Crossrefs

Cf. A118789, A118790, A032188.

Third column is A241765.

Sequence in context: A316152 A302204 A065077 * A196020 A346775 A028510

Adjacent sequences: A118785 A118786 A118787 * A118789 A118790 A118791

Keyword

nonn,tabl

Author

Paul D. Hanna, Apr 29 2006

Status

approved