A363916 - OEIS

A363916

Array read by descending antidiagonals. A(n, k) = Sum_{d=0..k} A363914(k, d) * n^d.

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 2, 3, 1, 0, 0, 6, 6, 4, 1, 0, 0, 12, 24, 12, 5, 1, 0, 0, 30, 72, 60, 20, 6, 1, 0, 0, 54, 240, 240, 120, 30, 7, 1, 0, 0, 126, 696, 1020, 600, 210, 42, 8, 1, 0, 0, 240, 2184, 4020, 3120, 1260, 336, 56, 9, 1

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

OFFSET

0,9

COMMENTS

Row n gives the number of n-ary sequences with primitive period k.

See A074650 and A143324 for combinatorial interpretations.

LINKS

Table of n, a(n) for n=0..65.

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

FORMULA

If k > 0 then k divides A(n, k), see the transposed array of A074650.

If k > 0 then n divides A(n, k), see the transposed array of A143325.

EXAMPLE

Array A(n, k) starts:

[0] 1, 0, 0, 0, 0, 0, 0, 0, 0, ... A000007

[1] 1, 1, 0, 0, 0, 0, 0, 0, 0, ... A019590

[2] 1, 2, 2, 6, 12, 30, 54, 126, 240, ... A027375

[3] 1, 3, 6, 24, 72, 240, 696, 2184, 6480, ... A054718

[4] 1, 4, 12, 60, 240, 1020, 4020, 16380, 65280, ... A054719

[5] 1, 5, 20, 120, 600, 3120, 15480, 78120, 390000, ... A054720

[6] 1, 6, 30, 210, 1260, 7770, 46410, 279930, 1678320, ... A054721

[7] 1, 7, 42, 336, 2352, 16800, 117264, 823536, 5762400, ... A218124

[8] 1, 8, 56, 504, 4032, 32760, 261576, 2097144, 16773120, ... A218125

A000012|A002378| A047928 | A218130 | A218131

A001477,A007531, A061167, A133499, (diagonal A252764)

Triangle T(n, k) starts:

[0] 1;

[1] 0, 1;

[2] 0, 1, 1;

[3] 0, 0, 2, 1;

[4] 0, 0, 2, 3, 1;

[5] 0, 0, 6, 6, 4, 1;

[6] 0, 0, 12, 24, 12, 5, 1;

[7] 0, 0, 30, 72, 60, 20, 6, 1;

[8] 0, 0, 54, 240, 240, 120, 30, 7, 1;

MAPLE

A363916 := (n, k) -> local d; add(A363914(k, d) * n^d, d = 0 ..k):

for n from 0 to 9 do seq(A363916(n, k), k = 0..8) od;

PROG

(SageMath)

def A363916(n, k): return sum(A363914(k, d) * n^d for d in range(k + 1))

for n in range(9): print([A363916(n, k) for k in srange(9)])

def T(n, k): return A363916(k, n - k)

CROSSREFS

Variant: A143324.

Rows: A000007 (n=0), A019590 (n=1), A027375 (n=2), A054718 (n=3), A054719 (n=4), A054720, A054721, A218124, A218125.

Columns: A000012 (k=0), A001477 (k=1), A002378 (k=2), A007531(k=3), A047928, A061167, A218130, A133499, A218131.

Cf. A252764 (main diagonal), A074650, A363914.

Sequence in context: A064301 A199881 A060701 * A275345 A259668 A261118

Adjacent sequences: A363913 A363914 A363915 * A363917 A363918 A363919

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Jul 04 2023

STATUS

approved