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

Offset: 0

Author

Peter Luschny, Jul 04 2023

Keywords

• nonn
• tabl

Comments

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

See A074650 and A143324 for combinatorial interpretations.

Examples

			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;
		

Links

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

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.

Programs

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;

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)

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.