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

%I A198204 #16 Oct 25 2015 17:34:36
%S A198204 1,1,2,1,9,12,1,28,120,120,1,75,750,2100,1680,1,186,3780,21840,45360,
%T A198204 30240,1,441,16856,176400,705600,1164240,665280,1,1016,69552,1224720,
%U A198204 8316000,25280640,34594560,17297280,1,2295,272250,7692300,82577880,408648240,998917920,1167566400,518918400
%N A198204 Series reversion of (1 - t*x)*log(1 + x) with respect to x.
%C A198204 This triangle is A133399 read by diagonals.
%F A198204 T(n,k) = k!*binomial(n + k - 1,k)*Stirling2(n,k + 1) (n >= 1, k >=0).
%F A198204 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 A198204 Main diagonal A001813, first subdiagonal A002691.
%F A198204 Column 1 A058877, column 2 A133386. Row sums A052892.
%F A198204 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 A198204 Triangle begins
%e A198204 .n\k.|..0....1.....2......3......4......5
%e A198204 = = = = = = = = = = = = = = = = = = = = =
%e A198204 ..1..|..1
%e A198204 ..2..|..1....2
%e A198204 ..3..|..1....9....12
%e A198204 ..4..|..1...28...120....120
%e A198204 ..5..|..1...75...750...2100...1680
%e A198204 ..6..|..1..186..3780..21840..45360..30240
%e A198204 ...
%t A198204 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 A198204 Cf. A001813, A002691, A052892, A058877, A133386, A133399, A141618.
%K A198204 nonn,easy,tabl
%O A198204 1,3
%A A198204 _Peter Bala_, Jul 31 2012