A198204 - OEIS

%I #16 Oct 25 2015 17:34:36

%S 1,1,2,1,9,12,1,28,120,120,1,75,750,2100,1680,1,186,3780,21840,45360,

%T 30240,1,441,16856,176400,705600,1164240,665280,1,1016,69552,1224720,

%U 8316000,25280640,34594560,17297280,1,2295,272250,7692300,82577880,408648240,998917920,1167566400,518918400

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

%C This triangle is A133399 read by diagonals.

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

%F 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! + ....

%F Main diagonal A001813, first subdiagonal A002691.

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

%F 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

%e Triangle begins

%e .n\k.|..0....1.....2......3......4......5

%e = = = = = = = = = = = = = = = = = = = = =

%e ..1..|..1

%e ..2..|..1....2

%e ..3..|..1....9....12

%e ..4..|..1...28...120....120

%e ..5..|..1...75...750...2100...1680

%e ..6..|..1..186..3780..21840..45360..30240

%e ...

%t 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 *)

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

%K nonn,easy,tabl

%O 1,3

%A _Peter Bala_, Jul 31 2012