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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. 3
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 (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 .

LINKS

Table of n, a(n) for n=0..48.

FORMULA

T(n,k) = (-1)^k*A075263(n,k). T(n,k)=(n-k)!*A008278(n+1,k+1).

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 ;...

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

CROSSREFS

Cf. A000142 A001710 A005460 A005461 A005462 A005463 A005464.

Sequence in context: A105613 A135894 A247500 * A075263 A130405 A058372

Adjacent sequences:  A130847 A130848 A130849 * A130851 A130852 A130853

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, Aug 20 2007

EXTENSIONS

New name by Peter Luschny, May 21 2013

STATUS

approved

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Last modified December 17 21:13 EST 2014. Contains 252040 sequences.