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

Author

Paul D. Hanna, Apr 29 2006

Keywords

• nonn
• tabl

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.

Examples

			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) +...
		

Crossrefs

Cf. A118789, A118790, A032188.

Third column is A241765.

Programs

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)}

Formula

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

Conjecture: T(n,k) = Sum_{j=0..k} binomial(n+j, n-k)*A269940(k, j) for 0 <= k <= n. - Mikhail Kurkov, Feb 17 2025