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

%I A154372 #38 Sep 08 2022 08:45:40
%S A154372 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,
%T A154372 375,36,1,1,448,5103,8960,4375,756,49,1,1,1024,20412,57344,43750,
%U A154372 12096,1372,64,1
%N A154372 Triangle T(n,k) = (k+1)^(n-k)*binomial(n,k).
%C A154372 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.
%C A154372 (*) 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.
%C A154372 The matrix inverse starts
%C A154372       1;
%C A154372      -1,      1;
%C A154372       3,     -4,     1;
%C A154372     -16,     24,    -9,     1;
%C A154372     125,   -200,    90,   -16,   1;
%C A154372   -1296,   2160, -1080,   240, -25,   1;
%C A154372   16807, -28812, 15435, -3920, 525, -36, 1;
%C A154372 .. compare with A122525 (row reversed). - _R. J. Mathar_, Mar 22 2013
%C A154372 From _Peter Bala_, Jan 14 2015: (Start)
%C A154372 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.
%C A154372 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.
%C A154372 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
%C A154372 /I_k 0\
%C A154372 \ 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)
%C A154372 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
%H A154372 G. C. Greubel, <a href="/A154372/b154372.txt">Table of n, a(n) for n = 0..1274</a>
%H A154372 Peter Bala, <a href="/A143395/a143395.pdf">A 3 parameter family of generalized Stirling numbers</a>
%H A154372 Emanuele Munarini, <a href="https://doi.org/10.2298/AADM180226017M">Combinatorial identities involving the central coefficients of a Sheffer matrix</a>, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.
%F A154372 T(n,k) = (k+1)^(n-k)*binomial(n,k). k!*T(n,k) gives the entries for A152818 read as a triangular array.
%F A154372 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
%F A154372 From _Peter Bala_, Jan 14 2015: (Start)
%F A154372 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.
%F A154372 Equals the matrix product A007318 * A059297. (End)
%e A154372 With the array M(k) as defined in the Comments section, the infinite product M(0)*M(1)*M(2)*... begins
%e A154372 /1      \ /1        \ /1        \      /1        \
%e A154372 |1 1     ||0 1       ||0 1      |      |1  1      |
%e A154372 |1 3 1   ||0 1 1     ||0 0 1    |... = |1  4  1   |
%e A154372 |1 6 5 1 ||0 1 3 1   ||0 0 1 1  |      |1 12  9  1|
%e A154372 |...     ||0 1 6 5 1 ||0 0 1 3 1|      |...       |
%e A154372 |...     ||...       ||...      |      |          |
%e A154372 - _Peter Bala_, Jan 13 2015
%t A154372 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 *)
%o A154372 (Magma) /* As triangle */ [[(k+1)^(n-k)*Binomial(n,k) : k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Sep 15 2016
%Y A154372 Cf. A080108, A152818, A059297, A145033.
%K A154372 nonn,easy,tabl
%O A154372 0,5
%A A154372 _Paul Curtz_, Jan 08 2009