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A137216 Erlang C queues type triangular sequence based on A122525.

%I A137216 #5 Jan 06 2022 15:41:54
%S A137216 1,1,1,2,2,3,6,9,22,41,24,64,266,708,1486,120,625,4536,17457,48088,
%T A137216 108129,720,7776,100392,563088,2043864,5709120,13399176,5040,117649,
%U A137216 2739472,22516209,107972560,375217945,1053757584,2544404617,40320,2097152,89020752,1076444064,6831882992,29566405440,99420254352,279663595232,688833593904
%N A137216 Erlang C queues type triangular sequence based on A122525.
%H A137216 G. C. Greubel, <a href="/A137216/b137216.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A137216 T(n, k) = (1/n)*( n^n * k^(n+1) - n! * (k - 1) * Sum_{j=0..n} (n*k)^j/j! ), with T(n, 0) = n! and T(n, 1) = n^(n-1).
%e A137216 Triangle begins as:
%e A137216     1;
%e A137216     1,    1;
%e A137216     2,    2,      3;
%e A137216     6,    9,     22,     41;
%e A137216    24,   64,    266,    708,    1486;
%e A137216   120,  625,   4536,  17457,   48088,  108129;
%e A137216   720, 7776, 100392, 563088, 2043864, 5709120, 13399176;
%t A137216 T[n_, k_]:= If[k==0, n!, If[k==1, n^(n-1), (1/n)*(k^(n+1)*n^n - n!*(k-1)*Sum[n^j*k^j/j!, {j,0,n}])]];
%t A137216 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 06 2022 *)
%o A137216 (Sage)
%o A137216 @CachedFunction
%o A137216 def A137216(n, k):
%o A137216     if (k==0): return factorial(n)
%o A137216     elif (k==1): return n^(n-1)
%o A137216     else: return (1/n)*(k^(n+1)*n^n - factorial(n)*(k-1)*sum((n*k)^j/factorial(j) for j in (0..n)))
%o A137216 flatten([[A137216(n, k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jan 06 2022
%Y A137216 Cf. A122525, A137227.
%K A137216 nonn,tabl
%O A137216 0,4
%A A137216 _Roger L. Bagula_, Mar 06 2008
%E A137216 Edited by _G. C. Greubel_, Jan 06 2022