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

0,17

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.

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

Example

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
...

Prog

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

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.

Sequence in context: A323011 A327747 A282750 * A226920 A362759 A123736

Adjacent sequences: A265887 A265888 A265889 * A265891 A265892 A265893

Keyword

nonn,tabl

Author

Antti Karttunen, Dec 19 2015

Status

approved