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A094310 Triangle read by rows: T(n,k), the k-th term of the n-th row, is the product of all numbers from 1 to n except k: T(n,k) = n!/k.

%I A094310 #41 Jun 08 2020 04:26:46
%S A094310 1,2,1,6,3,2,24,12,8,6,120,60,40,30,24,720,360,240,180,144,120,5040,
%T A094310 2520,1680,1260,1008,840,720,40320,20160,13440,10080,8064,6720,5760,
%U A094310 5040,362880,181440,120960,90720,72576,60480,51840,45360,40320,3628800,1814400,1209600,907200,725760,604800,518400,453600,403200,362880
%N A094310 Triangle read by rows: T(n,k), the k-th term of the n-th row, is the product of all numbers from 1 to n except k: T(n,k) = n!/k.
%C A094310 The sum of the rows gives A000254 (Stirling numbers of first kind). The first column and the leading diagonal are factorials given by A000142 with offsets of 0 and 1.
%C A094310 T(n,k) is the number of length k cycles in all permutations of {1..n}.
%C A094310 Second diagonal gives A001048(n). - _Anton Zakharov_, Oct 24 2016
%C A094310 T(n,k) is the number of permutations of [n] with all elements of [k] in a single cycle. To prove this result, let m denote the length of the cycle containing {1,..,k}. Letting m run from k to n, we obtain T(n,k) = Sum_{m=k..n} (C(n-k,m-k)*(m-1)!*(n-m)!) = n!/k. See an example below. - _Dennis P. Walsh_, May 24 2020
%H A094310 Alois P. Heinz, <a href="/A094310/b094310.txt">Rows n = 1..141, flattened</a>
%F A094310 E.g.f. for column k: x^k/(k*(1-x)).
%F A094310 T(n,k)*k = n*n! = A001563(n).
%e A094310 Triangle begins as:
%e A094310       1;
%e A094310       2,     1;
%e A094310       6,     3,     2;
%e A094310      24,    12,     8,     6;
%e A094310     120,    60,    40,    30,   24;
%e A094310     720,   360,   240,   180,  144,  120;
%e A094310    5040,  2520,  1680,  1260, 1008,  840,  720;
%e A094310   40320, 20160, 13440, 10080, 8064, 6720, 5760, 5040;
%e A094310   ...
%e A094310 T(4,2) counts the 12 permutations of [4] with elements 1 and 2 in the same cycle, namely, (1 2)(3 4), (1 2)(3)(4), (1 2 3)(4), (1 3 2)(4), (1 2 4)(3), (1 4 2)(3), (1 2 3 4), (1 2 4 3), (1 3 2 4), (1 3 4 2), (1 4 2 3), and (1 4 3 2). - _Dennis P. Walsh_, May 24 2020
%p A094310 seq(seq(n!/k, k=1..n), n=1..10);
%t A094310 Table[n!/k, {n,10}, {k,n}]//Flatten
%t A094310 Table[n!/Range[n], {n,10}]//Flatten (* _Harvey P. Dale_, Mar 12 2016 *)
%Y A094310 Cf. A000142, A000254, A001563, A001710, A002301.
%Y A094310 Cf. A061579, A094307, A110468, A133799.
%Y A094310 Cf. A129825. - _Johannes W. Meijer_, Jun 18 2009
%K A094310 nonn,tabl
%O A094310 1,2
%A A094310 _Amarnath Murthy_, Apr 29 2004
%E A094310 More terms from _Philippe Deléham_, Jun 11 2005