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

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

Author

Paul Curtz, Jan 08 2009

Keywords

• nonn
• easy
• tabl

Comments

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

Examples

			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
		

Links

• 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.

Crossrefs

Cf. A080108, A152818, A059297, A145033.

Programs

Magma

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

Mathematica

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 *)

Formula

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)