A265892 Array read by ascending antidiagonals: A(n,k) = A265893(A265609(n,k)), with n as row >= 0, k as column >= 0; the number of significant digits counted without trailing zeros 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, 2, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 2, 2, 2, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 0, 1, 3, 2, 3, 2, 2, 1, 1, 0, 1, 2, 3, 2, 2, 3, 1, 1, 1, 0, 1, 3, 1, 2, 3, 1, 2, 2, 1, 1, 0, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 0, 1, 3, 3, 4, 2, 2, 2, 3, 3, 2, 1, 1, 0, 1, 1, 3, 2, 3, 3, 3, 2, 2, 1, 1, 1, 1, 0, 1, 3, 3, 4, 3, 4, 4, 4, 3, 3, 2, 2, 1, 1, 0

Offset: 0

Author

Antti Karttunen, Dec 20 2015

Keywords

• nonn
• tabl
• base

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), ...

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
-
Counting such digits for each term, but without the trailing zeros gives us the top left corner of this array:
-
The top left corner of the array:
1, 0, 0, 0, 0, 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, 1, 1, 1, 1, 1, 1
1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1
1, 1, 2, 1, 2, 3, 2, 2, 3, 1, 2, 3, 2, 2, 3, 1, 2, 3, 2, 2, 3, 1, 2, 3, 2
1, 2, 2, 3, 2, 1, 2, 3, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 2, 3, 1, 2, 3, 4, 3
1, 1, 2, 2, 3, 1, 2, 2, 3, 4, 3, 1, 2, 3, 4, 2, 3, 2, 3, 4, 1, 2, 3, 3, 4
1, 3, 3, 2, 1, 2, 3, 4, 4, 4, 3, 4, 2, 3, 3, 4, 3, 4, 3, 3, 4, 2, 4, 5, 4
1, 2, 1, 1, 2, 3, 4, 3, 3, 2, 3, 2, 4, 5, 4, 3, 4, 3, 3, 4, 5, 3, 4, 3, 4
1, 3, 2, 4, 3, 4, 3, 4, 2, 3, 4, 5, 4, 3, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 3
1, 2, 3, 2, 3, 4, 3, 4, 5, 3, 4, 5, 3, 4, 5, 3, 4, 5, 5, 4, 5, 4, 5, 3, 4
1, 3, 3, 4, 4, 4, 3, 4, 4, 5, 4, 3, 3, 5, 6, 6, 5, 6, 5, 6, 5, 6, 4, 5, 6
1, 1, 3, 3, 3, 2, 3, 3, 4, 4, 5, 3, 4, 5, 5, 4, 5, 4, 5, 4, 5, 6, 4, 5, 4
1, 3, 4, 4, 4, 5, 4, 5, 5, 5, 5, 6, 4, 5, 6, 6, 5, 6, 5, 7, 6, 5, 5, 5, 5
1, 2, 3, 2, 4, 3, 4, 4, 4, 4, 5, 5, 6, 5, 5, 4, 6, 5, 6, 5, 4, 4, 4, 5, 6
1, 3, 1, 2, 3, 4, 5, 4, 3, 4, 4, 5, 5, 7, 6, 7, 6, 7, 5, 6, 7, 5, 4, 5, 6
1, 2, 4, 3, 5, 4, 3, 5, 6, 6, 5, 6, 6, 5, 6, 5, 6, 4, 5, 6, 4, 4, 6, 7, 8
1, 3, 3, 5, 4, 5, 5, 6, 5, 6, 5, 7, 6, 7, 6, 7, 4, 5, 6, 8, 5, 6, 7, 8, 6
1, 1, 3, 3, 4, 3, 5, 4, 5, 4, 6, 5, 6, 5, 6, 6, 7, 6, 7, 4, 5, 6, 7, 5, 6
...
		

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.

Row 0: A000007, rows 1-2: A000012, row 3: A000034 (see comment in A001710).

Column 0: A000012, column 1: A265893.

Cf. also array A265890.

Programs

Scheme

(define (A265892 n) (A265892bi (A025581 n) (A002262 n)))
(define (A265892bi row col) (A265893 (A265609bi row col)))

Formula

A(n,k) = A265893(A265609(n,k)).