Arie Harel has authored 2 sequences.
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).
Original entry on oeis.org
-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
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
A122525
Triangle read by rows: G(s, rho) = ((s-1)!/s)*Sum_{i=0..s-1} ((s-i)/i!)*(s*rho)^i.
Original entry on oeis.org
1, 1, 1, 2, 4, 3, 6, 18, 24, 16, 24, 96, 180, 200, 125, 120, 600, 1440, 2160, 2160, 1296, 720, 4320, 12600, 23520, 30870, 28812, 16807, 5040, 35280, 120960, 268800, 430080, 516096, 458752, 262144, 40320, 322560, 1270080, 3265920, 6123600, 8817984, 9920232, 8503056, 4782969
Offset: 1
G(5, rho) = 24 + 96*rho + 180*rho^2 + 200*rho^3 + 125*rho^4. The coefficients (24, 96, 180, 200, 125) give the 5th line of the triangle.
Triangle begins:
1;
1, 1;
2, 4, 3;
6, 18, 24, 16;
24, 96, 180, 200, 125;
120, 600, 1440, 2160, 2160, 1296;
720, 4320, 12600, 23520, 30870, 28812, 16807;
5040, 35280, 120960, 268800, 430080, 516096, 458752, 262144;
40320, 322560, 1270080, 3265920, 6123600, 8817984, 9920232, 8503056, 4782969;
- Cooper, R. B. 1981, Introduction to Queueing Theory. Second ed., North Holland, New York.
- Harel, A. 1988. Sharp Bounds and Simple Approximations for the Erlang Delay and Loss Formulas. Management Science, Vol. 34, 959-972.
- Harel, A. and P. Zipkin. 1987a. Strong Convexity Results for Queueing Systems. Operations Research, Vol. 35, No. 3, 405-418.
- Harel, A. and P. Zipkin. 1987b. The Convexity of a General Performance Measure for the Multi-Server Queues. Journal of Applied Probability, Vol. 24, 725-736.
- Jagers, A. A. and E. A. van Doorn, 1991. Convexity of functions which are generalizations of the Erlang loss function and the Erlang delay function. SIAM Review. Vol. 33 (2), 281-282.
- Lee, H. L. and M. A. Cohen. 1983. A Note on the Convexity of Performance Measures of M/M/c Queueing Systems. Journal of Applied Probability, Vol. 20, 920-923.
- Medhi, J. 2003. Stochastic Models in Queueing Theory. Second ed., Academic Press, New York.
- Smith, D.R. and W. Whitt. 1981. Resource Sharing for Efficiency in Traffic Systems. Bell System Technical Journal, Vol. 60, No. 1, 39-55.
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G:= proc(s) G(s):= (s-1)!/s*add((s-i)/i!*(s*rho)^i, i=0..(s-1)) end:
T:= n-> coeff(G(n), rho, k):
seq(seq(T(n, k), k=0..n-1), n=1..10); # Alois P. Heinz, Sep 08 2012
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(* First program *)
nn=6; t=Sum[n^(n-1)x^n/n!,{n,1,nn}]; f[list_]:=Select[list,#>0&]; Map[f,Map[Reverse,Range[0,nn]!CoefficientList[Series[Exp[t]/(1-y t),{x,0,nn}],{x,y}]]]//Grid (* Geoffrey Critzer, Sep 08 2012 *)
(* Second program *)
T[n_, k_]:= Coefficient[Series[((n-1)!/n)*Sum[(n-j)*(n*x)^j/j!, {j,0,n-1}], {x,0,30}], x, k];
Table[T[n, k], {n,10}, {k,0,n-1}]//Flatten (* G. C. Greubel, Jan 06 2022 *)
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def A122525(n,k): return ( (factorial(n-1)/n)*sum((n-j)*(n*x)^j/factorial(j) for j in (0..n-1)) ).series(x, n+1).list()[k]
flatten([[A122525(n,k) for k in (0..n-1)] for n in (1..12)]) # G. C. Greubel, Jan 06 2022
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