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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. 4
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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.

Triangle given by A123125*A007318 (as infinite lower triangular matrices), A123125 = Euler's triangle, A007318 = Pascal's triangle; A007318*A123125 gives A046802. Essentially reverse of A028246.

Taylor coefficients of Eulerian polynomials centered at 1. - Louis Zulli, Nov 28 2015

A signed refinement is A263634. - Tom Copeland, Nov 14 2016

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1034

FORMULA

T(n,k) = (-1)^k*A075263(n,k).

T(n,k) = (n-k)!*A008278(n+1,k+1).

T(n,n-1) = 2^n - 1 for n > 0. - Derek Orr, Dec 31 2015

E.g.f. is x/(e^(-xt)(1+x)-1). - Tom Copeland, Nov 14 2016

EXAMPLE

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

...

MATHEMATICA

Table[(n-k)!*Stirling2[n+1, n-k+1], {n, 0, 10}, {k, 0, n}] (* G. C. Greubel, Nov 15 2015 *)

PROG

(Sage)

def A130850(n, k):

    return add(EulerianNumber(n, j)*binomial(n-j, k) for j in (0..n))

for n in (0..7): [A130850(n, k) for k in (0..n)] # Peter Luschny, May 21 2013

(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

CROSSREFS

Cf. A000142, A001710, A005460, A005461, A005462, A005463, A005464.

Cf. A263634.

Sequence in context: A135894 A247500 A075263 * A130405 A058372 A128264

Adjacent sequences:  A130847 A130848 A130849 * A130851 A130852 A130853

KEYWORD

nonn,tabl,changed

AUTHOR

Philippe Deléham, Aug 20 2007

EXTENSIONS

New name from Peter Luschny, May 21 2013

STATUS

approved

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Last modified December 8 11:05 EST 2016. Contains 278939 sequences.