0,3

To get this sequence evaluate the following integral for r=1,2,3,4,...:

Int(-rho*exp(-rho*s*t)*t^r*s*ln(1+t),t=0..infinity);

for rho>0 and s>0. The evaluation yields two sequences.

For, example, when r=7, the above integral is equal to:

(exp(rho*s)*( - 2520*rho^2*s^2 - 5040 + 42*rho^5*s^5 + rho^7*s^7 - 210*rho^4*s^4 + 840*rho^3*s^3 + 5040*rho*s - 7*rho^6*s^6)*Ei(1,rho*s) - 13068 + 4320*rho*s - rho^6*s^6 - 51*rho^4*s^4 + 8*rho^5*s^5 + 272*rho^3*s^3 - 1200*rho^2*s^2)/(rho^7*s^7).

The first sequence

( - 2520*rho^2*s^2 - 5040 + 42*rho^5*s^5 + rho^7*s^7 - 210*rho^4*s^4 + 840*rho^3*s^3 + 5040*rho*s - 7*rho^6*s^6)

is known (A008279 or A068424?). The second sequence produces the current entry:

( - 13068 + 4320*rho*s - rho^6*s^6 - 51*rho^4*s^4 + 8*rho^5*s^5 + 272*rho^3*s^3 - 1200*rho^2*s^2),

whose coefficients

(-1, 8, -51, 272, -1200, 4320, -13068).

give the 7th line of the triangle.

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

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

Sequence in context: A025116 A081720 A137405 this_sequence A054631 A125077 A065253

Adjacent sequences: A121919 A121920 A121921 this_sequence A121923 A121924 A121925

sign,tabl

Arie Harel (Arie_Harel(AT)baruch.cuny.edu), Sep 09 2006