A145324 Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....

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

Offset: 1

Author

Jose Ramon Real, Oct 07 2008

Keywords

• nonn
• tabl

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

Examples

			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)
		

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]

Programs

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 *)

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

Extensions

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