A121922 The result of the integration Integral_{t=0..oo} -rho*exp(-rho*s*t)*t^j*s*log(1+t) dt can be written as (F(u,j)*exp(u)*Ei(1,u) + G(u,j))/u^j, where rho>0, s>0, and u=rho*s. Sequence is the regular triangle corresponding to G(u,j).

-1, 1, -3, -1, 4, -11, 1, -5, 18, -50, -1, 6, -27, 96, -274, 1, -7, 38, -168, 600, -1764, -1, 8, -51, 272, -1200, 4320, -13068, 1, -9, 66, -414, 2200, -9720, 35280, -109584, -1, 10, -83, 600, -3750, 19920, -88200, 322560, -1026576, 1, -11, 102, -836, 6024, -37620, 199920, -887040, 3265920, -10628640

Offset: 0

Author

Arie Harel, Sep 09 2006

Keywords

• sign
• tabl

Examples

			At j=7, the result of the integration Integral_{t=0..oo} -rho*exp(-rho*s*t)*t^j*s*log(1+t) dt
can be written as (F(u,7)*exp(u)*Ei(1,u) + G(u,7))/u^7, where
F(u,7) = u^7 - 7*u^6 + 42*u^5 - 210*u^4 + 840*u^3 -2520*u^2 + 5040*u - 5040,
G(u,7) = - u^6 + 8*u^5 - 51*u^4 + 272*u^3 - 1200*u^2 + 4320*u - 13068,
and u=rho*s.
The coefficients of F(u,7), i.e., (1, -7, 42, -210, 840, 2520, 5040, -5040), comprise the 7th row of A008279 (see also A068424). The coefficients of G(u,7), i.e., (-1, 8, -51, 272, -1200, 4320, -13068) give the 7th row of the triangle below.
Triangle begins:
  -1
  1, -3
  -1, 4, -11
  1, -5, 18, -50
  -1, 6, -27, 96, -274
  1, -7, 38, -168, 600, -1764
  -1, 8, -51, 272, -1200, 4320, -13068
		

Crossrefs

The right-hand diagonal is A000254, the one before that is A001563.

Cf. A008279, A068424.

Extensions

Edited by Jon E. Schoenfield, Oct 20 2013