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

1,3

Comments

This triangle is A133399 read by diagonals.

Links

Table of n, a(n) for n=1..45.

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

Example

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
...

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 *)

Crossrefs

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

Sequence in context: A063579 A240085 A078623 * A099599 A085488 A072265

Adjacent sequences: A198201 A198202 A198203 * A198205 A198206 A198207

Keyword

nonn,easy,tabl

Author

Peter Bala, Jul 31 2012

Status

approved