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%I #33 Sep 09 2020 17:47:56
%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
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