A122525 -id:A122525 - cp's OEIS frontend

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

Roger L. Bagula, Mar 06 2008

			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;

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

Cf. A122525, A137227.

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 *)

@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

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).

Edited by G. C. Greubel, Jan 06 2022

A066324 Number of endofunctions on n labeled points constructed from k rooted trees.

Original entry on oeis.org

1, 2, 2, 9, 12, 6, 64, 96, 72, 24, 625, 1000, 900, 480, 120, 7776, 12960, 12960, 8640, 3600, 720, 117649, 201684, 216090, 164640, 88200, 30240, 5040, 2097152, 3670016, 4128768, 3440640, 2150400, 967680, 282240, 40320, 43046721

Offset: 1

Christian G. Bower, Dec 14 2001

T(n,k) = number of endofunctions with k recurrent elements. - Mitch Harris, Jul 06 2006

The sum of row n is n^n, for any n. Basically the same sequence arises when studying random mappings (see A243203, A243202). - Stanislav Sykora, Jun 01 2014

			Triangle T(n,k) begins:
       1;
       2,      2;
       9,     12,      6;
      64,     96,     72,     24;
     625,   1000,    900,    480,   120;
    7776,  12960,  12960,   8640,  3600,   720;
  117649, 201684, 216090, 164640, 88200, 30240, 5040;
  ...

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 87, see (2.3.28).
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983, ex. 3.3.32.

Alois P. Heinz, Rows n = 1..141, flattened

Column 1: A000169.
Main diagonal: A000142.
T(n, n-1): A062119.
Row sums give A000312.
Cf. A001864, A021010, A063169, A122525, A144084, A243203.

T:= (n, k)-> k*n^(n-k)*(n-1)!/(n-k)!:
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Aug 22 2012

f[list_] := Select[list, # > 0 &]; t = Sum[n^(n - 1) x^n/n!, {n, 1, 20}]; Flatten[Map[f, Drop[Range[0, 10]! CoefficientList[Series[1/(1 - y*t), {x, 0, 10}], {x, y}], 1]]] (* Geoffrey Critzer, Dec 05 2011 *)

T(n, k)=k*n^(n-k)*(n-1)!/(n-k)! \\ Charles R Greathouse IV, Dec 05 2011

T(n,k) = k*n^(n-k)*(n-1)!/(n-k)!.
E.g.f. (relative to x): A(x, y)=1/(1-y*B(x)) - 1 = y*x +(2*y+2*y^2)*x^2/2! + (9*y+12*y^2+6*y^3)*x^3/3! + ..., where B(x) is e.g.f. A000169.
From Peter Bala, Sep 30 2011: (Start)
Let F(x,t) = x/(1+t*x)*exp(-x/(1+t*x)) = x*(1 - (1+t)*x + (1+4*t+2*t^2)*x^2/2! - ...). F is essentially the e.g.f. for A144084 (see also A021010). Then the e.g.f. for the present table is t*F(x,t)^(-1), where the compositional inverse is taken with respect to x.
Removing a factor of n from the n-th row entries results in A122525 in row reversed form.
(End)
Sum_{k=2..n} (k-1) * T(n,k) = A001864(n). - Geoffrey Critzer, Aug 19 2013
Sum_{k=1..n} k * T(n,k) = A063169(n). - Alois P. Heinz, Dec 15 2021

A154372 Triangle T(n,k) = (k+1)^(n-k)*binomial(n,k).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 12, 9, 1, 1, 32, 54, 16, 1, 1, 80, 270, 160, 25, 1, 1, 192, 1215, 1280, 375, 36, 1, 1, 448, 5103, 8960, 4375, 756, 49, 1, 1, 1024, 20412, 57344, 43750, 12096, 1372, 64, 1

Offset: 0

Paul Curtz, Jan 08 2009

From A152650/A152656,coefficients of other exponential polynomials(*). a(n) is triangle A152818 where terms of each column is divided by the beginning one. See A000004, A001787(n+1), A006043=2*A027472, A006044=6*A038846.
(*) Not factorial as written in A006044. See A000110, Bell-Touchard. Second diagonal is 1,4,9,16,25, denominators of Lyman's spectrum of hydrogen, A000290(n+1) which has homogeneous indices for denominators series of Rydberg-Ritz spectrum of hydrogen.
The matrix inverse starts
      1;
     -1,      1;
      3,     -4,     1;
    -16,     24,    -9,     1;
    125,   -200,    90,   -16,   1;
  -1296,   2160, -1080,   240, -25,   1;
  16807, -28812, 15435, -3920, 525, -36, 1;
.. compare with A122525 (row reversed). - R. J. Mathar, Mar 22 2013
From Peter Bala, Jan 14 2015: (Start)
Exponential Riordan array [exp(z), z*exp(z)]. This triangle is the particular case a = 0, b = 1, c = 1 of the triangle of generalized Stirling numbers of the second kind S(a,b,c) defined in the Bala link. Cf. A059297.
This is the triangle of connection constants when expressing the monomials x^n as a linear combination of the basis polynomials (x - 1)*(x - k - 1)^(k-1), k = 0,1,2,.... For example, from row 3 we have x^3 = 1 + 12*(x - 1) + 9*(x - 1)*(x - 3) + (x - 1)*(x - 4)^2.
Let M be the infinite lower unit triangular array with (n,k)-th entry (k*(n - k + 1) + 1)/(k + 1)*binomial(n,k). M is the row reverse of A145033. For k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/ having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to the present triangle. See the Example section. (End)
T(n,k) is also the number of idempotent partial transformations of {1,2,...,n} having exactly k fixed points. - Geoffrey Critzer, Nov 25 2021

			With the array M(k) as defined in the Comments section, the infinite product M(0)*M(1)*M(2)*... begins
/1      \ /1        \ /1        \      /1        \
|1 1     ||0 1       ||0 1      |      |1  1      |
|1 3 1   ||0 1 1     ||0 0 1    |... = |1  4  1   |
|1 6 5 1 ||0 1 3 1   ||0 0 1 1  |      |1 12  9  1|
|...     ||0 1 6 5 1 ||0 0 1 3 1|      |...       |
|...     ||...       ||...      |      |          |
- _Peter Bala_, Jan 13 2015

G. C. Greubel, Table of n, a(n) for n = 0..1274
Peter Bala, A 3 parameter family of generalized Stirling numbers
Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.

Cf. A080108, A152818, A059297, A145033.

/* As triangle */ [[(k+1)^(n-k)*Binomial(n,k) : k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 15 2016

T[n_, k_] := (k + 1)^(n - k)*Binomial[n, k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Sep 15 2016 *)

T(n,k) = (k+1)^(n-k)*binomial(n,k). k!*T(n,k) gives the entries for A152818 read as a triangular array.
E.g.f.: exp(x*(1+t*exp(x))) = 1 + (1+t)*x + (1+4*t+t^2)*x^2/2! + (1+12*t+9*t^2+t*3)*x^3/3! + .... O.g.f.: Sum_{k>=1} (t*x)^(k-1)/(1-k*x)^k = 1 + (1+t)*x + (1+4*t+t^2)*x^2 + .... Row sums are A080108. - Peter Bala, Oct 09 2011
From Peter Bala, Jan 14 2015: (Start)
Recurrence equation: T(n+1,k+1) = T(n,k+1) + Sum_{j = 0..n-k} (j + 1)*binomial(n,j)*T(n-j,k) with T(n,0) = 1 for all n.
Equals the matrix product A007318 * A059297. (End)

A137227 Triangle T(n, k) = n^(n-1) * Fibonacci(k)^(n+1) - (n-1)! * (Fibonacci(k) - 1) * Sum_{j=0..n} (n*Fibonacci(k))^j/j!, with T(n, 0) = n! and T(n, 1) = n^(n-1), read by rows.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 6, 9, 9, 22, 24, 64, 64, 266, 708, 120, 625, 625, 4536, 17457, 108129, 720, 7776, 7776, 100392, 563088, 5709120, 52517688, 5040, 117649, 117649, 2739472, 22516209, 375217945, 5489293264, 92757410569, 40320, 2097152, 2097152, 89020752, 1076444064, 29566405440, 688833593904, 18867973329344, 513683908057152

Offset: 0

Roger L. Bagula, Mar 07 2008

			Triangle begins as:
     1;
     1,      1;
     2,      2,      2;
     6,      9,      9,      22;
    24,     64,     64,     266,      708;
   120,    625,    625,    4536,    17457,    108129;
   720,   7776,   7776,  100392,   563088,   5709120,   52517688;
  5040, 117649, 117649, 2739472, 22516209, 375217945, 5489293264, 92757410569;

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

Cf. A000045, A122525, A137216.

T[n_, k_]:= If[k==0, n!, If[k==1, n^(n-1), (1/n)*(Fibonacci[k]^(n+1)*n^n - n!*(Fibonacci[k] -1)*Sum[n^j*Fibonacci[k]^j/j!, {j,0,n}])]];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jan 06 2022 *)

@CachedFunction
def A137227(n,k):
    if (k==0): return factorial(n)
    elif (k==1): return n^(n-1)
    else: return (1/n)*(fibonacci(k)^(n+1)*n^n - factorial(n)*(fibonacci(k) -1)*sum((n*fibonacci(k))^j/factorial(j) for j in (0..n)))
flatten([[A137227(n,k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Jan 06 2022

T(n, k) = (1/n)*( n^n * Fibonacci(k)^(n+1) - n! * (Fibonacci(k) - 1) * Sum_{j=0..n} (n*Fibonacci(k))^j/j! ), with T(n, 0) = n! and T(n, 1) = n^(n-1).

Edited by G. C. Greubel, Jan 06 2022

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A137216 Erlang C queues type triangular sequence based on A122525.

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A066324 Number of endofunctions on n labeled points constructed from k rooted trees.

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A154372 Triangle T(n,k) = (k+1)^(n-k)*binomial(n,k).

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A137227 Triangle T(n, k) = n^(n-1) * Fibonacci(k)^(n+1) - (n-1)! * (Fibonacci(k) - 1) * Sum_{j=0..n} (n*Fibonacci(k))^j/j!, with T(n, 0) = n! and T(n, 1) = n^(n-1), read by rows.

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