A265890 Array read by ascending antidiagonals: A(n,k) = A099563(A265609(n,k)), with n as row >= 0, k as column >= 0; the most significant digit in the factorial base representation of rising factorial n^(k) = (n+k-1)!/(n-1)!.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 2, 3, 2, 1, 1, 0, 1, 1, 1, 1, 3, 1, 1, 0, 1, 1, 1, 1, 1, 3, 1, 1, 0, 1, 1, 2, 2, 2, 1, 4, 1, 1, 0, 1, 1, 3, 4, 4, 3, 1, 4, 1, 1, 0, 1, 1, 3, 1, 1, 6, 3, 1, 5, 1, 1, 0, 1, 1, 4, 1, 1, 1, 8, 4, 1, 5, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 1, 5, 2, 6, 1, 1, 0, 1, 2, 1, 2, 3, 3, 3, 2, 1, 6, 2, 6, 1, 1, 0

Offset: 0

Author

Antti Karttunen, Dec 19 2015

Keywords

• nonn
• tabl

Comments

Square array A(row,col) is read by ascending antidiagonals as: A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), A(3,0), A(2,1), A(1,2), A(0,3), ...

A265609(n,k) is the rising factorial, also known as Pochhammer symbol and A099563(n) is the most significant "digit" (place holder) in the factorial representation (A007623) of n.

Examples

			The top left corner of the array A265609 with its terms shown in factorial base (A007623) looks like this:
1,   0,    0,     0,       0,        0,         0,          0,           0
1,   1,   10,   100,    1000,    10000,    100000,    1000000,    10000000
1,  10,  100,  1000,   10000,   100000,   1000000,   10000000,   100000000
1,  11,  200,  2200,   30000,   330000,   4000000,   44000000,   500000000
1,  20,  310, 10000,  110000,  1220000,  14000000,  160000000,  1830000000
1,  21, 1100, 13300,  220000,  3000000,  36000000,  452000000,  5500000000
1, 100, 1300, 24000,  411000,  6000000,  82000000, 1100000000, 13300000000
1, 101, 2110, 41000, 1000000, 13000000, 174000000, 2374000000, 30360000000
-
Taking the most significant "digit" (placeholder that may get arbitrarily large values) gives us the top left corner of this array:
-
1, 0, 0, 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
1, 1, 1, 1, 1, 1, 1, 1,  1, 1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1
1, 1, 1, 1, 1, 1, 1, 1,  1, 1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1
1, 1, 2, 2, 3, 3, 4, 4,  5, 5,  6,  6,  7,  7,  8,  8,  9,  9, 10, 10, 11
1, 2, 3, 1, 1, 1, 1, 1,  1, 2,  2,  2,  2,  2,  2,  3,  3,  3,  3,  3,  3
1, 2, 1, 1, 2, 3, 3, 4,  5, 6,  7,  8, 10, 11, 12, 14, 15, 17, 19, 21,  1
1, 1, 1, 2, 4, 6, 8, 1,  1, 1,  1,  2,  2,  2,  2,  3,  3,  3,  4,  4,  5
1, 1, 2, 4, 1, 1, 1, 2,  3, 3,  4,  5,  6,  8,  9, 11, 12, 14, 16, 19, 21
1, 1, 3, 1, 1, 2, 3, 4,  6, 8, 11, 14,  1,  1,  1,  1,  2,  2,  2,  3,  3
1, 1, 3, 1, 2, 3, 5, 8,  1, 1,  1,  2,  2,  3,  4,  5,  6,  7,  8, 10, 12
1, 1, 4, 1, 3, 5, 9, 1,  2, 2,  3,  5,  6,  8, 11, 14, 17, 21,  1,  1,  1
1, 1, 1, 2, 4, 8, 1, 2,  3, 5,  7, 10, 14,  1,  1,  1,  2,  2,  3,  3,  4
1, 2, 1, 3, 6, 1, 2, 4,  6, 9, 14,  1,  1,  2,  3,  4,  5,  6,  8, 10, 13
1, 2, 1, 3, 1, 2, 3, 6, 10, 1,  1,  2,  3,  5,  6,  9, 12, 16, 21,  1,  1
1, 2, 1, 4, 1, 2, 5, 9,  1, 2,  3,  4,  7, 10, 14, 20,  1,  1,  2,  2,  3
1, 2, 2, 5, 1, 3, 7, 1,  2, 3,  5,  8, 13,  1,  1,  1,  2,  3,  4,  6,  8
...
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..7259; the first 120 antidiagonals of array
• Index entries for sequences related to factorial base representation

Crossrefs

Cf. A007623, A265609.

Column 1: A099563.

Row 0: A000007, rows 1 & 2: A000012, row 3: A008619 (see comment in A001710).

Row 4: 1,2,3 followed by A097992 ?

Main diagonal: A265891 (essentially, without the initial 1 from the corner of this array).

Cf. also array A265892.

Programs

Scheme

(define (A265890 n) (A265890bi (A025581 n) (A002262 n)))
(define (A265890bi row col) (A099563 (A265609bi row col))) ;; Code for A265609bi given in A265609.