A265892 - OEIS

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)!.

%I #17 Jun 03 2018 02:02:41

%S 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,

%T 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,

%U 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

%N 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)!.

%C 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), ...

%H Antti Karttunen, <a href="/A265892/b265892.txt">Table of n, a(n) for n = 0..7259; the first 120 antidiagonals of array</a>

%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>

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

%e The top left corner of the array A265609 with its terms shown in factorial base (A007623) looks like this:

%e 1, 0, 0, 0, 0, 0, 0, 0, 0

%e 1, 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000

%e 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000

%e 1, 11, 200, 2200, 30000, 330000, 4000000, 44000000, 500000000

%e 1, 20, 310, 10000, 110000, 1220000, 14000000, 160000000, 1830000000

%e 1, 21, 1100, 13300, 220000, 3000000, 36000000, 452000000, 5500000000

%e 1, 100, 1300, 24000, 411000, 6000000, 82000000, 1100000000, 13300000000

%e 1, 101, 2110, 41000, 1000000, 13000000, 174000000, 2374000000, 30360000000

%e -

%e Counting such digits for each term, but without the trailing zeros gives us the top left corner of this array:

%e -

%e The top left corner of the array:

%e 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

%e 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

%e 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

%e 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

%e 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

%e 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

%e 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

%e 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

%e 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

%e 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

%e 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

%e 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

%e 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

%e 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

%e 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

%e 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

%e 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

%e 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

%e ...

%o (Scheme)

%o (define (A265892 n) (A265892bi (A025581 n) (A002262 n)))

%o (define (A265892bi row col) (A265893 (A265609bi row col)))

%Y Cf. A007623, A265609.

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

%Y Column 0: A000012, column 1: A265893.

%Y Cf. also array A265890.

%K nonn,tabl,base

%O 0,12

%A _Antti Karttunen_, Dec 20 2015