OFFSET
0,2
COMMENTS
Note that the table includes the well-known sequence (A000165) discussed by Gordon on pages 636-645 of AMM 106 (1999).
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)
EXAMPLE
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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alford Arnold, Sep 06 2004
STATUS
approved