A276617 -id:A276617

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A276955

Square array A(row,col): A(row,1) = A273670(row-1), and for col > 1, A(row,col) = A153880(A(row,col-1)); Dispersion of factorial base left shift A153880.

+10
20

1, 2, 3, 6, 8, 4, 24, 30, 12, 5, 120, 144, 48, 14, 7, 720, 840, 240, 54, 26, 9, 5040, 5760, 1440, 264, 126, 32, 10, 40320, 45360, 10080, 1560, 744, 150, 36, 11, 362880, 403200, 80640, 10800, 5160, 864, 168, 38, 13, 3628800, 3991680, 725760, 85680, 41040, 5880, 960, 174, 50, 15, 39916800, 43545600, 7257600, 766080, 367920, 46080, 6480, 984, 246, 56, 16

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

The square array A(row,col) is read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

When viewed in factorial base (A007623) the terms on each row start all with the same prefix, but with an increasing number of zeros appended to the end. For example, for row 8 (A001344 from a(1)=11 onward), the terms written in factorial base look as: 121, 1210, 12100, 121000, ...

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..1830; the first 60 antidiagonals of array

Index entries for sequences related to factorial base representation

Index entries for sequences that are permutations of the natural numbers

FORMULA

A(row,1) = A273670(row-1), and for col > 1, A(row,col) = A153880(A(row,col-1))

As a composition of other permutations:

a(n) = A275848(A257505(n)).

EXAMPLE

The top left {1..9} x {1..18} corner of the array:

1, 2, 6, 24, 120, 720, 5040, 40320, 362880

3, 8, 30, 144, 840, 5760, 45360, 403200, 3991680

4, 12, 48, 240, 1440, 10080, 80640, 725760, 7257600

5, 14, 54, 264, 1560, 10800, 85680, 766080, 7620480

7, 26, 126, 744, 5160, 41040, 367920, 3669120, 40279680

9, 32, 150, 864, 5880, 46080, 408240, 4032000, 43908480

10, 36, 168, 960, 6480, 50400, 443520, 4354560, 47174400

11, 38, 174, 984, 6600, 51120, 448560, 4394880, 47537280

13, 50, 246, 1464, 10200, 81360, 730800, 7297920, 80196480

15, 56, 270, 1584, 10920, 86400, 771120, 7660800, 83825280

16, 60, 288, 1680, 11520, 90720, 806400, 7983360, 87091200

17, 62, 294, 1704, 11640, 91440, 811440, 8023680, 87454080

18, 72, 360, 2160, 15120, 120960, 1088640, 10886400, 119750400

19, 74, 366, 2184, 15240, 121680, 1093680, 10926720, 120113280

20, 78, 384, 2280, 15840, 126000, 1128960, 11249280, 123379200

21, 80, 390, 2304, 15960, 126720, 1134000, 11289600, 123742080

22, 84, 408, 2400, 16560, 131040, 1169280, 11612160, 127008000

23, 86, 414, 2424, 16680, 131760, 1174320, 11652480, 127370880

PROG

(Scheme)

(define (A276955 n) (A276955bi (A002260 n) (A004736 n)))

(define (A276955bi row col) (if (= 1 col) (A273670 (- row 1)) (A153880 (A276955bi row (- col 1)))))

CROSSREFS

Inverse permutation: A276956.

Transpose: A276953.

Cf. A276949 (index of column where n appears), A276951 (index of row).

Cf. A153880.

Columns 1-3: A273670, A276932, A276933.

The following lists some of the rows that have their own entries. Pattern present in the factorial base expansion of the terms on that row is given in double quotes:

Row 1: A000142 (from a(1)=1, "1" onward),

Row 2: A001048 (from a(2)=3, "11" onward),

Row 3: A052849 (from a(2)=4, "20" onward).

Row 4: A052649 (from a(1)=5, "21" onward).

Row 5: A108217 (from a(3)=7, "101" onward).

Row 6: A054119 (from a(3)=9, "111" onward).

Row 7: A052572 (from a(2)=10, "120" onward).

Row 8: A001344 (from a(1)=11, "121" onward).

Row 13: A052560 (from a(3)=18, "300" onward).

Row 16: A225658 (from a(1)=21, "311" onward).

Row 20: A276940 (from a(3) = 27, "1011" onward).

Related or similar permutations: A257505, A275848, A273666.

Cf. also arrays A276617, A276588 & A276945.

KEYWORD

nonn,base,tabl

AUTHOR

Antti Karttunen, Sep 22 2016

STATUS

approved

A286625

Square array A(n,k) = A276945(n,k)/A002110(k-1), read by descending antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

+10
9

1, 1, 3, 1, 4, 4, 1, 6, 6, 5, 1, 8, 10, 7, 7, 1, 12, 14, 11, 16, 9, 1, 14, 22, 15, 36, 19, 10, 1, 18, 26, 23, 78, 41, 21, 11, 1, 20, 34, 27, 144, 85, 45, 22, 13, 1, 24, 38, 35, 222, 155, 91, 46, 31, 15, 1, 30, 46, 39, 324, 235, 165, 92, 71, 34, 16, 1, 32, 58, 47, 438, 341, 247, 166, 155, 76, 36, 17, 1, 38, 62, 59, 668, 457, 357, 248, 287, 162, 80, 37, 18

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..1275; the first 50 antidiagonals of array

Index entries for sequences related to primorial base

FORMULA

A(n,k) = A276945(n, k) / A002110(k-1).

EXAMPLE

The top left 12 X 12 corner of the array:

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38

4, 6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74

5, 7, 11, 15, 23, 27, 35, 39, 47, 59, 63, 75

7, 16, 36, 78, 144, 222, 324, 438, 668, 900, 1148, 1518

9, 19, 41, 85, 155, 235, 341, 457, 691, 929, 1179, 1555

10, 21, 45, 91, 165, 247, 357, 475, 713, 957, 1209, 1591

11, 22, 46, 92, 166, 248, 358, 476, 714, 958, 1210, 1592

13, 31, 71, 155, 287, 443, 647, 875, 1335, 1799, 2295, 3035

15, 34, 76, 162, 298, 456, 664, 894, 1358, 1828, 2326, 3072

16, 36, 80, 168, 308, 468, 680, 912, 1380, 1856, 2356, 3108

17, 37, 81, 169, 309, 469, 681, 913, 1381, 1857, 2357, 3109

PROG

(Scheme)

(define (A286625 n) (A286625bi (A002260 n) (A004736 n)))

(define (A286625bi row col) (/ (A276945bi row col) (A002110 (- col 1))))

CROSSREFS

Transpose: A286623.

Cf. A002110, A276945.

Column 1: A276155.

Row 1: A000012, Row 2: A008864, Row 3: A100484, Row 4: A072055, Row 5: A023523 (from its second term onward), Row 6: A286624.

Cf. A276617 (analogous array).

KEYWORD

nonn,tabl

AUTHOR

Antti Karttunen, Jun 28 2017

STATUS

approved

A276616

Square array A(n,k) = A276953(n,k)/n!, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

+10
6

1, 3, 1, 4, 4, 1, 5, 6, 5, 1, 7, 7, 8, 6, 1, 9, 13, 9, 10, 7, 1, 10, 16, 21, 11, 12, 8, 1, 11, 18, 25, 31, 13, 14, 9, 1, 13, 19, 28, 36, 43, 15, 16, 10, 1, 15, 25, 29, 40, 49, 57, 17, 18, 11, 1, 16, 28, 41, 41, 54, 64, 73, 19, 20, 12, 1, 17, 30, 45, 61, 55, 70, 81, 91, 21, 22, 13, 1, 18, 31, 48, 66, 85, 71, 88, 100, 111, 23, 24, 14, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of array

Index entries for sequences related to factorial base representation

FORMULA

A(n,k) = A276953(n,k)/n!

EXAMPLE

The top left corner of the array:

1, 3, 4, 5, 7, 9, 10, 11, 13, 15, 16, 17, 18, 19, 20, 21, 22

1, 4, 6, 7, 13, 16, 18, 19, 25, 28, 30, 31, 36, 37, 39, 40, 42

1, 5, 8, 9, 21, 25, 28, 29, 41, 45, 48, 49, 60, 61, 64, 65, 68

1, 6, 10, 11, 31, 36, 40, 41, 61, 66, 70, 71, 90, 91, 95, 96, 100

1, 7, 12, 13, 43, 49, 54, 55, 85, 91, 96, 97, 126, 127, 132, 133, 138

1, 8, 14, 15, 57, 64, 70, 71, 113, 120, 126, 127, 168, 169, 175, 176, 182

PROG

(Scheme)

(define (A276616 n) (A276616bi (A002260 n) (A004736 n)))

(define (A276616bi row col) (/ (A276953bi row col) (A000142 row)))

CROSSREFS

Transpose: A276617.

Cf. A000142, A276953.

Row 1: A273670, row 2: A276931, row 3: A276934.

For columns, see the rows of transpose A276617.

KEYWORD

nonn,base,tabl

AUTHOR

Antti Karttunen, Sep 22 2016

STATUS

approved

A276934

Row 3 of A276616: a(n) = A276933(n) / 6.

+10
5

1, 5, 8, 9, 21, 25, 28, 29, 41, 45, 48, 49, 60, 61, 64, 65, 68, 69, 121, 125, 128, 129, 141, 145, 148, 149, 161, 165, 168, 169, 180, 181, 184, 185, 188, 189, 241, 245, 248, 249, 261, 265, 268, 269, 281, 285, 288, 289, 300, 301, 304, 305, 308, 309, 361, 365, 368, 369, 381, 385, 388, 389, 401, 405, 408, 409, 420, 421, 424

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Starting offset is 0 (with a(0) = 2) to match with the starting offset of A276933 and A273670.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..10000

Index entries for sequences related to factorial base representation

FORMULA

a(n) = A276933(n) / 6.

PROG

(Scheme) (define (A276934 n) (/ (A276933 n) 6))

CROSSREFS

Cf. A273670, A276933.

Row 3 of A276616, column 3 of A276617.

KEYWORD

nonn,base

AUTHOR

Antti Karttunen, Sep 23 2016

STATUS

approved

A276931

Row 2 of A276616: a(n) = A276932(n)/2.

+10
4

1, 4, 6, 7, 13, 16, 18, 19, 25, 28, 30, 31, 36, 37, 39, 40, 42, 43, 61, 64, 66, 67, 73, 76, 78, 79, 85, 88, 90, 91, 96, 97, 99, 100, 102, 103, 121, 124, 126, 127, 133, 136, 138, 139, 145, 148, 150, 151, 156, 157, 159, 160, 162, 163, 181, 184, 186, 187, 193, 196, 198, 199, 205, 208, 210, 211, 216, 217, 219, 220, 222

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Starting offset is 0 (with a(0) = 1) to match with the starting offset of A276932 and A273670.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..10000

Index entries for sequences related to factorial base representation

FORMULA

a(n) = A276932(n)/2.

PROG

(define (A276931 n) (/ (A276932 n) 2))