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

0,4

Links

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

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).

Example

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;

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 *)

Prog

(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

Crossrefs

Cf. A122525, A137227.

Sequence in context: A163493 A321405 A054200 * A369424 A070550 A298179

Adjacent sequences: A137213 A137214 A137215 * A137217 A137218 A137219

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 06 2008

Extensions

Edited by G. C. Greubel, Jan 06 2022

Status

approved