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

0,2

Comments

Note that the table includes the well-known sequence (A000165) discussed by Gordon on pages 636-645 of AMM 106 (1999).

Links

Table of n, a(n) for n=0..20.

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

Cf. A000165, A052876, A098348, A098385, A194649.

Sequence in context: A060149 A376467 A059374 * A243253 A064536 A361456

Adjacent sequences: A098381 A098382 A098383 * A098385 A098386 A098387

Keyword

nonn,tabl,more

Author

Alford Arnold, Sep 06 2004

Status

approved