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
Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150, flattened)
Olivier Bodini, Antoine Genitrini, Mehdi Naima, Ranked Schröder Trees, arXiv:1808.08376 [cs.DS], 2018.
Olivier Bodini, Antoine Genitrini, Cécile Mailler, Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
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
T(n,k) = T(n-1,k)+n*T(n-1,k-1). - Mikhail Kurkov, Jun 26 2018
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
MATHEMATICA
Table[Reverse[CoefficientList[Product[x+j, {j, 2, k}], x]], {k, 1, 15}] // Flatten (* Robert A. Russell, Sep 29 2018 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Jose Ramon Real, Oct 07 2008
EXTENSIONS
More terms from R. J. Mathar, Oct 10 2008
STATUS
approved