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%I #16 Sep 11 2017 15:53:13
%S 1,2,1,3,4,2,4,9,12,6,5,16,36,48,24,6,25,80,180,240,120,7,36,150,480,
%T 1080,1440,720,8,49,252,1050,3360,7560,10080,5040,9,64,392,2016,8400,
%U 26880,60480,80640,40320,10,81,576,3528,18144,75600,241920,544320
%N A factorial-like triangle read by rows: T(0,0) = 1; T(n+1,0) = T(n,0)+1; T(n+1,k+1) = T(n,0)*T(n,k), k=0..n.
%C row(0) = {1}; row(n+1) = row(n) multiplied by n and prepended with (n+1);
%C A111063(n+1) = sum of n-th row;
%C T(2*n,n) = A002690(n), central terms;
%C T(n,0) = n + 1;
%C T(n,1) = A000290(n), n > 0;
%C T(n,2) = A011379(n-1), n > 1;
%C T(n,3) = A047927(n), n > 2;
%C T(n,4) = A192849(n-1), n > 3;
%C T(n,5) = A000142(5) * A027810(n-5), n > 4;
%C T(n,6) = A000142(6) * A027818(n-6), n > 5;
%C T(n,7) = A000142(7) * A056001(n-7), n > 6;
%C T(n,8) = A000142(8) * A056003(n-8), n > 7;
%C T(n,9) = A000142(9) * A056114(n-9), n > 8;
%C T(n,n-10) = 11 * A051431(n-10), n > 9;
%C T(n,n-9) = 10 * A049398(n-9), n > 8;
%C T(n,n-8) = 9 * A049389(n-8), n > 7;
%C T(n,n-7) = 8 * A049388(n-7), n > 6;
%C T(n,n-6) = 7 * A001730(n), n > 5;
%C T(n,n-5) = 6 * A001725(n), n > 5;
%C T(n,n-4) = 5 * A001720(n), n > 4;
%C T(n,n-3) = 4 * A001715(n), n > 2;
%C T(n,n-2) = A070960(n), n > 1;
%C T(n,n-1) = A052849(n), n > 0;
%C T(n,n) = A000142(n);
%C T(n,k) = A137948(n,k) * A007318(n,k), 0 <= k <= n.
%H Reinhard Zumkeller, <a href="/A245334/b245334.txt">Rows n = 0..125 of triangle, flattened</a>
%F T(n,k) = n!*(n+1-k)/(n-k)!. - _Werner Schulte_, Sep 09 2017
%e . 0: 1;
%e . 1: 2, 1;
%e . 2: 3, 4, 2;
%e . 3: 4, 9, 12, 6;
%e . 4: 5, 16, 36, 48, 24;
%e . 5: 6, 25, 80, 180, 240, 120;
%e . 6: 7, 36, 150, 480, 1080, 1440, 720;
%e . 7: 8, 49, 252, 1050, 3360, 7560, 10080, 5040;
%e . 8: 9, 64, 392, 2016, 8400, 26880, 60480, 80640, 40320;
%e . 9: 10, 81, 576, 3528, 18144, 75600, 241920, 544320, 725760, 362880.
%t Table[(n!)/((n - k)!)*(n + 1 - k), {n, 0, 9}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Sep 10 2017 *)
%o (Haskell)
%o a245334 n k = a245334_tabl !! n !! k
%o a245334_row n = a245334_tabl !! n
%o a245334_tabl = iterate (\row@(h:_) -> (h + 1) : map (* h) row) [1]
%Y Cf. A111063 (row sums), A240993 (row products), A002690 (central terms).
%Y Cf. A000290, A011379, A027810, A027818, A047927, A056001, A056003, A056114, A192849.
%Y Cf. A000142, A001715, A001720, A001725, A001730, A049388, A049389, A049398, A051431, A052849, A070960.
%Y Cf. A007318, A137948.
%K nonn,tabl
%O 0,2
%A _Reinhard Zumkeller_, Aug 30 2014
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