A198204 Series reversion of (1 - t*x)*log(1 + x) with respect to x.

1, 1, 2, 1, 9, 12, 1, 28, 120, 120, 1, 75, 750, 2100, 1680, 1, 186, 3780, 21840, 45360, 30240, 1, 441, 16856, 176400, 705600, 1164240, 665280, 1, 1016, 69552, 1224720, 8316000, 25280640, 34594560, 17297280, 1, 2295, 272250, 7692300, 82577880, 408648240, 998917920, 1167566400, 518918400

Offset: 1

Author

Peter Bala, Jul 31 2012

Keywords

• nonn
• easy
• tabl

Comments

This triangle is A133399 read by diagonals.

Examples

			Triangle begins
.n\k.|..0....1.....2......3......4......5
= = = = = = = = = = = = = = = = = = = = =
..1..|..1
..2..|..1....2
..3..|..1....9....12
..4..|..1...28...120....120
..5..|..1...75...750...2100...1680
..6..|..1..186..3780..21840..45360..30240
...
		

Crossrefs

Cf. A001813, A002691, A052892, A058877, A133386, A133399, A141618.

Programs

Mathematica

Flatten[CoefficientList[CoefficientList[InverseSeries[Series[Log[1 + x]*(1 - t*x),{x,0,9}]], x]*Table[n!, {n,0,9}], t]] (* Peter Luschny, Oct 25 2015 *)

Formula

T(n,k) = k!*binomial(n + k - 1,k)*Stirling2(n,k + 1) (n >= 1, k >=0).

E.g.f.: A(x,t) = series reversion of (1 - t*x)*log(1 + x) w.r.t. x = x + (1 + 2*t)*x^2/2! + (1 + 9*t + 12*t^2)*x^3/3! + ....

Main diagonal A001813, first subdiagonal A002691.

Column 1 A058877, column 2 A133386. Row sums A052892.

1 - t*A(x,t) = x/series reversion of x*(1 - t(exp(x) - 1)) with respect to x. Cf. A141618. - Peter Bala, Oct 22 2015