Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #40 Apr 27 2024 03:34:39
%S 1,1,2,1,5,6,1,9,26,24,1,14,71,154,120,1,20,155,580,1044,720,1,27,295,
%T 1665,5104,8028,5040,1,35,511,4025,18424,48860,69264,40320,1,44,826,
%U 8624,54649,214676,509004,663696,362880,1,54,1266,16884,140889,761166
%N Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....
%C The last number of row n is n!.
%C 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
%C 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
%H Michael De Vlieger, <a href="/A145324/b145324.txt">Table of n, a(n) for n = 1..11325</a> (rows 1 <= n <= 150, flattened)
%H Olivier Bodini, Antoine Genitrini, Mehdi Naima, <a href="https://arxiv.org/abs/1808.08376">Ranked Schröder Trees</a>, arXiv:1808.08376 [cs.DS], 2018.
%H Olivier Bodini, Antoine Genitrini, Cécile Mailler, Mehdi Naima, <a href="https://hal.archives-ouvertes.fr/hal-02865198">Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study</a>, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
%H Robert E. Moritz, <a href="/A001701/a001701.pdf">On the sum of products of n consecutive integers</a>, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy]
%F T(n,k) = A143491(n+1,n+2-k). - _R. J. Mathar_, Oct 10 2008
%F 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
%F T(n,k) = T(n-1,k)+n*T(n-1,k-1). - _Mikhail Kurkov_, Jun 26 2018
%e From _Wolfdieter Lang_, Oct 24 2011: (Start)
%e n\k 1 2 3 4 5 6 7 ...
%e 1: 1
%e 2: 1 2
%e 3: 1 5 6
%e 4: 1 9 26 24
%e 5: 1 14 71 154 120
%e 6: 1 20 155 580 1044 720
%e 7: 1 27 295 1665 5104 8028 5040
%e ...
%e T(4,3)= 26 = |s(5,3)| - |s(5,4)| + |s(5,5)| = 35 - 10 + 1.
%e (End)
%p 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
%t Table[Reverse[CoefficientList[Product[x+j, {j, 2, k}], x]], {k, 1, 15}] // Flatten (* _Robert A. Russell_, Sep 29 2018 *)
%K nonn,tabl
%O 1,3
%A _Jose Ramon Real_, Oct 07 2008
%E More terms from _R. J. Mathar_, Oct 10 2008