A098384 Triangle read by rows of coefficients used to generate diagonals of ordered factorizations as displayed in A098348.
1, 3, 2, 13, 18, 8, 75, 158, 144, 48, 541, 1530, 2120, 1440, 384, 4683, 16622, 30960, 31920, 17280, 3840
Offset: 0
Examples
The table begins: 1 3 2 13 18 8 75 158 144 48 541 1530 2120 1440 384 The binomial transform of (13,18,8) yields 13,31,57,91,... The binomial transform of 13,31,57,91,... yields 13,44,132,368,... A098385
Formula
From Peter Bala, Apr 20 2012: (Start)
The following formulas are all conjectural:
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).
Recurrence equation:
T(n,k) = 2*k*T(n-1,k-1) + 3*(k+1)*T(n-1,k) + (k+2)*T(n-1,k+1).
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! + ....
Column n generating function: 2^n*exp(x)*(1-exp(x))^n/(exp(x)-2)^(n+2) for n >= 0.
(End)
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