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A363916
Array read by descending antidiagonals. A(n, k) = Sum_{d=0..k} A363914(k, d) * n^d.
1
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
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
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
.
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
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Jul 04 2023
STATUS
approved