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A145324 Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); .... 8
1, 1, 2, 1, 5, 6, 1, 9, 26, 24, 1, 14, 71, 154, 120, 1, 20, 155, 580, 1044, 720, 1, 27, 295, 1665, 5104, 8028, 5040, 1, 35, 511, 4025, 18424, 48860, 69264, 40320, 1, 44, 826, 8624, 54649, 214676, 509004, 663696, 362880, 1, 54, 1266, 16884, 140889, 761166 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The last number of row n is n!.

Essentially the triangle given by [1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [2, 1, 3, 2, 4, 3, 5, 4, 6, 5, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 09 2008

T(n+1,k+1) = a_k(2,3,...,n+1), n >= 0, k = 0..n, with the elementary symmetric function a_k(x[1],x[2],...,x[n]),

  with a_0(0):=1. E.g., a_2(2,3,4) = 2*3 + 2*4 + 3*4 = 26 = T(4,3). - Wolfdieter Lang, Oct 24 2011

LINKS

Table of n, a(n) for n=1..51.

Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy]

FORMULA

T(n,k) = A143491(n+1,n+2-k). - R. J. Mathar, Oct 10 2008

T(n,k) = Sum_{m=0..k-1} (-1)^m*|s(n+1, n+2-k+m)|, n >= 1, k = 1..n, with the Stirling numbers of the first kind s(n,k) = A048994(n,k). - Wolfdieter Lang, Oct 24 2011

EXAMPLE

From Wolfdieter Lang, Oct 24 2011: (Start)

n\k 1  2   3    4     5    6     7 ...

1:  1

2:  1  2

3:  1  5   6

4:  1  9  26   24

5:  1 14  71  154   120

6:  1 20 155  580  1044  720

7:  1 27 295 1665  5104 8028  5040

...

T(4,3)= 26 = |s(5,3)| - |s(5,4)| + |s(5,5)| = 35 - 10 + 1.

(End)

MAPLE

A145324 := proc(n, k) coeftayl( 1*mul(x+i, i=2..n), x=0, n-k) ; end: for n from 1 to 11 do for k from 1 to n do printf("%d, ", A145324(n, k)) ; od: od: # R. J. Mathar, Oct 10 2008

CROSSREFS

Sequence in context: A193722 A193635 A241168 * A260613 A179457 A107783

Adjacent sequences:  A145321 A145322 A145323 * A145325 A145326 A145327

KEYWORD

nonn,tabl

AUTHOR

Jose Ramon Real, Oct 07 2008

EXTENSIONS

More terms from R. J. Mathar, Oct 10 2008

STATUS

approved

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Last modified February 22 09:44 EST 2018. Contains 299449 sequences. (Running on oeis4.)