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A154372 Triangle T(n,k) = (k+1)^(n-k)*binomial(n,k). 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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)

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1274

P. Bala, A 3 parameter family of generalized Stirling numbers

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..inf} (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)

EXAMPLE

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

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 *)

PROG

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

CROSSREFS

Cf. A080108, A152818, A059297, A145033.

Sequence in context: A205946 A101919 A055106 * A080416 A213166 A168619

Adjacent sequences:  A154369 A154370 A154371 * A154373 A154374 A154375

KEYWORD

nonn,easy,tabl

AUTHOR

Paul Curtz, Jan 08 2009

STATUS

approved

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Last modified November 20 18:16 EST 2019. Contains 329337 sequences. (Running on oeis4.)