A137216 Erlang C queues type triangular sequence based on A122525.

1, 1, 1, 2, 2, 3, 6, 9, 22, 41, 24, 64, 266, 708, 1486, 120, 625, 4536, 17457, 48088, 108129, 720, 7776, 100392, 563088, 2043864, 5709120, 13399176, 5040, 117649, 2739472, 22516209, 107972560, 375217945, 1053757584, 2544404617, 40320, 2097152, 89020752, 1076444064, 6831882992, 29566405440, 99420254352, 279663595232, 688833593904

Offset: 0

Author

Roger L. Bagula, Mar 06 2008

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
    1;
    1,    1;
    2,    2,      3;
    6,    9,     22,     41;
   24,   64,    266,    708,    1486;
  120,  625,   4536,  17457,   48088,  108129;
  720, 7776, 100392, 563088, 2043864, 5709120, 13399176;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A122525, A137227.

Programs

Mathematica

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}])]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jan 06 2022 *)

Sage

@CachedFunction
def A137216(n, k):
    if (k==0): return factorial(n)
    elif (k==1): return n^(n-1)
    else: return (1/n)*(k^(n+1)*n^n - factorial(n)*(k-1)*sum((n*k)^j/factorial(j) for j in (0..n)))
flatten([[A137216(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jan 06 2022

Formula

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).

Extensions

Edited by G. C. Greubel, Jan 06 2022