A245334 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.

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, 1080, 1440, 720, 8, 49, 252, 1050, 3360, 7560, 10080, 5040, 9, 64, 392, 2016, 8400, 26880, 60480, 80640, 40320, 10, 81, 576, 3528, 18144, 75600, 241920, 544320

Offset: 0

Author

Reinhard Zumkeller, Aug 30 2014

Keywords

• nonn
• tabl

Comments

row(0) = {1}; row(n+1) = row(n) multiplied by n and prepended with (n+1);

A111063(n+1) = sum of n-th row;

T(2*n,n) = A002690(n), central terms;

T(n,0) = n + 1;

T(n,1) = A000290(n), n > 0;

T(n,2) = A011379(n-1), n > 1;

T(n,3) = A047927(n), n > 2;

T(n,4) = A192849(n-1), n > 3;

T(n,5) = A000142(5) * A027810(n-5), n > 4;

T(n,6) = A000142(6) * A027818(n-6), n > 5;

T(n,7) = A000142(7) * A056001(n-7), n > 6;

T(n,8) = A000142(8) * A056003(n-8), n > 7;

T(n,9) = A000142(9) * A056114(n-9), n > 8;

T(n,n-10) = 11 * A051431(n-10), n > 9;

T(n,n-9) = 10 * A049398(n-9), n > 8;

T(n,n-8) = 9 * A049389(n-8), n > 7;

T(n,n-7) = 8 * A049388(n-7), n > 6;

T(n,n-6) = 7 * A001730(n), n > 5;

T(n,n-5) = 6 * A001725(n), n > 5;

T(n,n-4) = 5 * A001720(n), n > 4;

T(n,n-3) = 4 * A001715(n), n > 2;

T(n,n-2) = A070960(n), n > 1;

T(n,n-1) = A052849(n), n > 0;

T(n,n) = A000142(n);

T(n,k) = A137948(n,k) * A007318(n,k), 0 <= k <= n.

Examples

			.  0:   1;
.  1:   2,  1;
.  2:   3,  4,   2;
.  3:   4,  9,  12,    6;
.  4:   5, 16,  36,   48,    24;
.  5:   6, 25,  80,  180,   240,   120;
.  6:   7, 36, 150,  480,  1080,  1440,    720;
.  7:   8, 49, 252, 1050,  3360,  7560,  10080,   5040;
.  8:   9, 64, 392, 2016,  8400, 26880,  60480,  80640,  40320;
.  9:  10, 81, 576, 3528, 18144, 75600, 241920, 544320, 725760, 362880.
		

Links

• Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Crossrefs

Cf. A111063 (row sums), A240993 (row products), A002690 (central terms).

Cf. A000290, A011379, A027810, A027818, A047927, A056001, A056003, A056114, A192849.

Cf. A000142, A001715, A001720, A001725, A001730, A049388, A049389, A049398, A051431, A052849, A070960.

Cf. A007318, A137948.

Programs

Haskell

a245334 n k = a245334_tabl !! n !! k
a245334_row n = a245334_tabl !! n
a245334_tabl = iterate (\row@(h:_) -> (h + 1) : map (* h) row) [1]

Mathematica

Table[(n!)/((n - k)!)*(n + 1 - k), {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Sep 10 2017 *)

Formula

T(n,k) = n!*(n+1-k)/(n-k)!. - Werner Schulte, Sep 09 2017