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A130850 Triangle read by rows, 0 <= k <= n, T(n,k) = Sum_{j=0..n} A(n,j)*binomial(n-j,k) where A(n,j) are the Eulerian numbers A173018.

%I A130850 #86 Nov 10 2021 20:35:56
%S A130850 1,1,1,2,3,1,6,12,7,1,24,60,50,15,1,120,360,390,180,31,1,720,2520,
%T A130850 3360,2100,602,63,1,5040,20160,31920,25200,10206,1932,127,1,40320,
%U A130850 181440,332640,317520,166824,46620,6050,255,1,362880,1814400,3780000,4233600,2739240,1020600,204630,18660,511,1
%N A130850 Triangle read by rows, 0 <= k <= n, T(n,k) = Sum_{j=0..n} A(n,j)*binomial(n-j,k) where A(n,j) are the Eulerian numbers A173018.
%C A130850 Old name was: Triangle T(n,k), 0<=k<=n, read by rows given by [1,1,2,2,3,3,4,4,5,5,...] DELTA [1,0,2,0,3,0,4,0,5,0,6,0,...] where DELTA is the operator defined in A084938.
%C A130850 Vandervelde (2018) refers to this as the Worpitzky number triangle - _N. J. A. Sloane_, Mar 27 2018 [Named after the German mathematician Julius Daniel Theodor Worpitzky (1835-1895). - _Amiram Eldar_, Jun 24 2021]
%C A130850 Triangle given by A123125*A007318 (as infinite lower triangular matrices), A123125 = Euler's triangle, A007318 = Pascal's triangle; A007318*A123125 gives A046802.
%C A130850 Taylor coefficients of Eulerian polynomials centered at 1. - _Louis Zulli_, Nov 28 2015
%C A130850 A signed refinement is A263634. - _Tom Copeland_, Nov 14 2016
%C A130850 With all offsets 0, let A_n(x;y) = (y + E.(x))^n, an Appell sequence in y where E.(x)^k = E_k(x) are the Eulerian polynomials of A123125. Then the row polynomials of A046802 (the h-polynomials of the stellahedra) are given by h_n(x) = A_n(x;1); the row polynomials of A248727 (the face polynomials of the stellahedra), by f_n(x) = A_n(1 + x;1); the Swiss-knife polynomials of A119879, by Sw_n(x) = A_n(-1;1 + x); and the row polynomials of this entry (the Worpitsky triangle, A130850), by w_n(x) = A(1 + x;0). Other specializations of A_n(x;y) give A090582 (the f-polynomials of the permutohedra, cf. also A019538) and A028246 (another version of the Worpitsky triangle). - _Tom Copeland_, Jan 24 2020
%H A130850 G. C. Greubel, <a href="/A130850/b130850.txt">Table of n, a(n) for n = 0..1034</a> [a(438) and a(901) corrected by _Georg Fischer_, Nov 10 2021]
%H A130850 Nguyen-Huu-Bong, <a href="https://doi.org/10.1016/0097-3165(71)90049-5">Some Combinatorial Properties of Summation Operators</a>, J. Comb. Theory, Ser. A, Vo. 11, No. 3 (1971), pp. 213-221. See Table on page 214.
%H A130850 John K. Sikora, <a href="https://arxiv.org/abs/1806.00887">On Calculating the Coefficients of a Polynomial Generated Sequence Using the Worpitzky Number Triangles</a>, arXiv:1806.00887 [math.NT], 2018.
%H A130850 Sam Vandervelde, <a href="https://doi.org/10.1080/00029890.2018.1408347">The Worpitzky Numbers Revisited</a>, Amer. Math. Monthly, Vol. 125, No. 3 (2018), pp. 198-206.
%F A130850 T(n,k) = (-1)^k*A075263(n,k).
%F A130850 T(n,k) = (n-k)!*A008278(n+1,k+1).
%F A130850 T(n,n-1) = 2^n - 1 for n > 0. - _Derek Orr_, Dec 31 2015
%F A130850 E.g.f.: x/(e^(-x*t)*(1+x)-1). - _Tom Copeland_, Nov 14 2016
%F A130850 Sum_{k=1..floor(n/2)} T(n,2k) = Sum_{k=0..floor(n/2)} T(n,2k+1) = A000670(n). - _Jacob Sprittulla_, Oct 03 2021
%e A130850 Triangle begins:
%e A130850 1
%e A130850 1      1
%e A130850 2      3       1
%e A130850 6      12      7       1
%e A130850 24     60      50      15      1
%e A130850 120    360     390     180     31      1
%e A130850 720    2520    3360    2100    602     63      1
%e A130850 5040   20160   31920   25200   10206   1932    127    1
%e A130850 40320  181440  332640  317520  166824  46620   6050   255   1
%e A130850 362880 1814400 3780000 4233600 2739240 1020600 204630 18660 511 1
%e A130850 ...
%t A130850 Table[(n-k)!*StirlingS2[n+1, n-k+1], {n, 0, 10}, {k, 0, n}] (* _G. C. Greubel_, Nov 15 2015 *)
%o A130850 (Sage)
%o A130850 from sage.combinat.combinat import eulerian_number
%o A130850 def A130850(n, k):
%o A130850     return add(eulerian_number(n, j)*binomial(n-j, k) for j in (0..n))
%o A130850 for n in (0..7): [A130850(n, k) for k in (0..n)] # _Peter Luschny_, May 21 2013
%o A130850 (PARI) t(n, k) = (n-k)!*stirling(n+1, n-k+1, 2);
%o A130850 tabl(nn) = for (n=0, 10, for (k=0, n, print1(t(n,k),", ")); print()); \\ _Michel Marcus_, Nov 16 2015
%Y A130850 Essentially reverse of A028246.
%Y A130850 Variants are A130850, A075263, A163626.
%Y A130850 Cf. A000142, A001710, A005460, A005461, A005462, A005463, A005464, A263634.
%Y A130850 Cf. A019538, A028246, A046802, A090582, A119879, A123125, A248727.
%K A130850 nonn,tabl
%O A130850 0,4
%A A130850 _Philippe Deléham_, Aug 20 2007
%E A130850 New name from _Peter Luschny_, May 21 2013