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A154372 Triangle T(n,k) = (k+1)^(n-k)*binomial(n,k). 4
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)
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
LINKS
Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.
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)
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
Sequence in context: A205946 A101919 A055106 * A080416 A213166 A168619
KEYWORD
nonn,easy,tabl
AUTHOR
Paul Curtz, Jan 08 2009
STATUS
approved

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Last modified March 28 11:59 EDT 2024. Contains 371254 sequences. (Running on oeis4.)