This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A122525 #63 May 05 2025 04:40:03
%S A122525 1,1,1,2,4,3,6,18,24,16,24,96,180,200,125,120,600,1440,2160,2160,1296,
%T A122525 720,4320,12600,23520,30870,28812,16807,5040,35280,120960,268800,
%U A122525 430080,516096,458752,262144,40320,322560,1270080,3265920,6123600,8817984,9920232,8503056,4782969
%N A122525 Triangle read by rows: G(s, rho) = ((s-1)!/s)*Sum_{i=0..s-1} ((s-i)/i!)*(s*rho)^i.
%C A122525 When s is a positive integer and 0 < rho < 1 then C(s,rho):=(s*rho)^s/G(s,rho)/s is the well-known Erlang delay (or the Erlang's C) formula. This measure is a basic formula of queueing theory. The applications of this formula are in diverse systems where queueing phenomena arise, including telecommunications, production, and service systems. The formula gives the steady-state probability of delay in the M/M/s queueing system. The number of servers is denoted by s and the traffic intensity is denoted by rho, 0 < rho < 1, where rho=(arrival rate)/(service rate)/s.
%C A122525 With offset = 0, T(n,n-k) is the number of partial functions on {1,2,...,n} with exactly k recurrent elements for 0 <= k <= n. Row sums = (n+1)^n. - _Geoffrey Critzer_, Sep 08 2012
%D A122525 Cooper, R. B. 1981, Introduction to Queueing Theory. Second ed., North Holland, New York.
%D A122525 Harel, A. 1988. Sharp Bounds and Simple Approximations for the Erlang Delay and Loss Formulas. Management Science, Vol. 34, 959-972.
%D A122525 Harel, A. and P. Zipkin. 1987a. Strong Convexity Results for Queueing Systems. Operations Research, Vol. 35, No. 3, 405-418.
%D A122525 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.
%D A122525 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.
%D A122525 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.
%D A122525 Medhi, J. 2003. Stochastic Models in Queueing Theory. Second ed., Academic Press, New York.
%D A122525 Smith, D.R. and W. Whitt. 1981. Resource Sharing for Efficiency in Traffic Systems. Bell System Technical Journal, Vol. 60, No. 1, 39-55.
%H A122525 Alois P. Heinz, <a href="/A122525/b122525.txt">Rows n = 1..141, flattened</a>
%H A122525 C. W. J. Beenakker, <a href="https://arxiv.org/abs/2501.11085">Entropy and singular-value moments of products of truncated random unitary matrices</a>, arXiv:2501.11085 [quant-ph], 2025. See p. 6.
%H A122525 E. Brockmeyer, H. L. Halstrøm and Arne Jensen, <a href="https://web.archive.org/web/20120207182954/http://oldwww.com.dtu.dk/teletraffic/Erlang.html">The Life and Works of A.K. Erlang</a>
%F A122525 An equivalent expression for G(s, rho) that is often used is: G(s, rho) = (1-rho)*(s-1)!*Sum_{i=0..s-1} (s^i*rho^i/i!) + rho^s*s^(s-1).
%F A122525 For s > 0 and rho > 0 one can use the expression: G(s, rho) = (exp(s*rho)*s*rho*(1-rho)*(s-1)*Gamma(s-1, s*rho) + rho^s*s^s)/(s*rho).
%F A122525 For s > 0 and rho > 0 one can also use the integral representation G(s, rho) = ((s*rho)^s/s)*Integral_{t=0..oo} rho*s*exp(-rho*s*t)*t*(1+t)^(s-1) dt.
%F A122525 Multiplying the n-th row entries by n+1 results in triangle A066324 in row reversed form. - _Peter Bala_, Sep 30 2011
%F A122525 Row generating polynomials are given by (1/n)*D^n(1/(1-x*t)) evaluated at x = 0, where D is the operator exp(x)/(1-x)*d/dx. - _Peter Bala_, Dec 27 2011
%e A122525 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.
%e A122525 Triangle begins:
%e A122525 1;
%e A122525 1, 1;
%e A122525 2, 4, 3;
%e A122525 6, 18, 24, 16;
%e A122525 24, 96, 180, 200, 125;
%e A122525 120, 600, 1440, 2160, 2160, 1296;
%e A122525 720, 4320, 12600, 23520, 30870, 28812, 16807;
%e A122525 5040, 35280, 120960, 268800, 430080, 516096, 458752, 262144;
%e A122525 40320, 322560, 1270080, 3265920, 6123600, 8817984, 9920232, 8503056, 4782969;
%p A122525 G:= proc(s) G(s):= (s-1)!/s*add((s-i)/i!*(s*rho)^i, i=0..(s-1)) end:
%p A122525 T:= n-> coeff(G(n), rho, k):
%p A122525 seq(seq(T(n, k), k=0..n-1), n=1..10); # _Alois P. Heinz_, Sep 08 2012
%t A122525 (* First program *)
%t A122525 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 *)
%t A122525 (* Second program *)
%t A122525 T[n_, k_]:= Coefficient[Series[((n-1)!/n)*Sum[(n-j)*(n*x)^j/j!, {j,0,n-1}], {x,0,30}], x, k];
%t A122525 Table[T[n, k], {n,10}, {k,0,n-1}]//Flatten (* _G. C. Greubel_, Jan 06 2022 *)
%o A122525 (Sage)
%o A122525 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]
%o A122525 flatten([[A122525(n,k) for k in (0..n-1)] for n in (1..12)]) # _G. C. Greubel_, Jan 06 2022
%Y A122525 Cf. A066324, A137216, A137227.
%K A122525 nonn,tabl
%O A122525 1,4
%A A122525 _Arie Harel_, Sep 14 2006