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

0,2

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.

Links

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

Formula

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

Example

.  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.

Mathematica

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

Prog

(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]

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.

Sequence in context: A128544 A120058 A208532 * A102756 A086614 A108959

Adjacent sequences: A245331 A245332 A245333 * A245335 A245336 A245337

Keyword

nonn,tabl

Author

Reinhard Zumkeller, Aug 30 2014

Status

approved