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A098384 Triangle read by rows of coefficients used to generate diagonals of ordered factorizations as displayed in A098348.

%I A098384 #5 Apr 20 2012 11:28:56
%S A098384 1,3,2,13,18,8,75,158,144,48,541,1530,2120,1440,384,4683,16622,30960,
%T A098384 31920,17280,3840
%N A098384 Triangle read by rows of coefficients used to generate diagonals of ordered factorizations as displayed in A098348.
%C A098384 Note that the table includes the well-known sequence (A000165) discussed by Gordon on pages 636-645 of AMM 106 (1999).
%F A098384 From Peter Bala, Apr 20 2012: (Start)
%F A098384 The following formulas are all conjectural:
%F A098384 T(n,k) = 2^k*sum {i = k+1..n+1} binomial(i,k+1)*(i-1)!*Stirling2(n+1,i) = 1/(k+1)*A194649(n+1,k).
%F A098384 Recurrence equation:
%F A098384 T(n,k) = 2*k*T(n-1,k-1) + 3*(k+1)*T(n-1,k) + (k+2)*T(n-1,k+1).
%F A098384 E.g.f.: exp(x)/((2-exp(x))*(2*t+2-(2*t+1)*exp(x))) = 1 + (3+2*t)*x + (13+18*t+8*t^2)*x^2/2! + ....
%F A098384 Column n generating function: 2^n*exp(x)*(1-exp(x))^n/(exp(x)-2)^(n+2) for n >= 0.
%F A098384 (End)
%e A098384 The table begins:
%e A098384 1
%e A098384 3 2
%e A098384 13 18 8
%e A098384 75 158 144 48
%e A098384 541 1530 2120 1440 384
%e A098384 The binomial transform of (13,18,8) yields 13,31,57,91,...
%e A098384 The binomial transform of 13,31,57,91,... yields 13,44,132,368,... A098385
%Y A098384 Cf. A000165, A052876, A098348, A098385. A194649.
%K A098384 nonn,tabl
%O A098384 0,2
%A A098384 _Alford Arnold_, Sep 06 2004