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A130850
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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.
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9
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1, 1, 1, 2, 3, 1, 6, 12, 7, 1, 24, 60, 50, 15, 1, 120, 360, 390, 180, 31, 1, 720, 2520, 3360, 2100, 602, 63, 1, 5040, 20160, 31920, 25200, 10206, 1932, 127, 1, 40320, 181440, 332640, 317520, 166824, 46620, 6050, 255, 1, 362880, 1814400, 3780000, 4233600, 2739240, 1020600, 204630, 18660, 511, 1
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OFFSET
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0,4
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COMMENTS
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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.
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]
Taylor coefficients of Eulerian polynomials centered at 1. - Louis Zulli, Nov 28 2015
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
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LINKS
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FORMULA
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T(n,n-1) = 2^n - 1 for n > 0. - Derek Orr, Dec 31 2015
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EXAMPLE
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Triangle begins:
1
1 1
2 3 1
6 12 7 1
24 60 50 15 1
120 360 390 180 31 1
720 2520 3360 2100 602 63 1
5040 20160 31920 25200 10206 1932 127 1
40320 181440 332640 317520 166824 46620 6050 255 1
362880 1814400 3780000 4233600 2739240 1020600 204630 18660 511 1
...
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MATHEMATICA
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Table[(n-k)!*StirlingS2[n+1, n-k+1], {n, 0, 10}, {k, 0, n}] (* G. C. Greubel, Nov 15 2015 *)
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PROG
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(Sage)
from sage.combinat.combinat import eulerian_number
return add(eulerian_number(n, j)*binomial(n-j, k) for j in (0..n))
(PARI) t(n, k) = (n-k)!*stirling(n+1, n-k+1, 2);
tabl(nn) = for (n=0, 10, for (k=0, n, print1(t(n, k), ", ")); print()); \\ Michel Marcus, Nov 16 2015
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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