A074650 -id:A074650 - cp's OEIS frontend

A006180 Witt vector *5!/5!.

1, 472, 467133, 636430764, 1038934571875, 1903882757758426, 3782689379194538057, 7975541699963490241566, 17602442746255160006062232, 40278440105728693363331297293

Offset: 1

Simon Plouffe

nonn

If c is the Witt transform of b then b(n) = Sum_{d|n} A074650(n/d, c(d)).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

H. Gaudier, Relèvement des coefficients binomiaux dans les vecteurs de Witt, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 16 (1988), 358/S-18, pp. 93-108.
H. Gaudier, Relèvement des coefficients binomiaux dans les vecteurs de Witt, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 16 (1988), 358/S-18, pp. 93-107. (Annotated scanned copy)

Witt transform of A074654.

More terms and formula from Christian G. Bower, Aug 28 2002

A363916 Array read by descending antidiagonals. A(n, k) = Sum_{d=0..k} A363914(k, d) * n^d.

Original entry on oeis.org

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

Peter Luschny, Jul 04 2023

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

See A074650 and A143324 for combinatorial interpretations.

			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;

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

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.

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;

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)

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.

A383023 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) is the n-th term of the inverse Weigh transform of j-> k^j.

Original entry on oeis.org

1, 2, 1, 3, 3, 0, 4, 6, 2, 1, 5, 10, 8, 6, 0, 6, 15, 20, 24, 6, 0, 7, 21, 40, 70, 48, 11, 0, 8, 28, 70, 165, 204, 124, 18, 1, 9, 36, 112, 336, 624, 690, 312, 36, 0, 10, 45, 168, 616, 1554, 2620, 2340, 834, 56, 0, 11, 55, 240, 1044, 3360, 7805, 11160, 8230, 2184, 105, 0

Offset: 1

Seiichi Manyama, Apr 12 2025

			Square array begins:
  1,  2,   3,    4,     5,     6,      7, ...
  1,  3,   6,   10,    15,    21,     28, ...
  0,  2,   8,   20,    40,    70,    112, ...
  1,  6,  24,   70,   165,   336,    616, ...
  0,  6,  48,  204,   624,  1554,   3360, ...
  0, 11, 124,  690,  2620,  7805,  19656, ...
  0, 18, 312, 2340, 11160, 39990, 117648, ...

Christian G. Bower, PARI programs for transforms, 2007.
N. J. A. Sloane, Maple programs for transforms, 2001-2020.

Columns k=1..5 give A209229, A306156, A306157, A306158, A306159.

Cf. A074650.

A(n,k) = (1/n) * (k^n + Sum_{d

Product_{n>=1} (1 + x^n)^A(n,k) = 1/(1 - k*x).

A347277 Table T(n,k) read by downward antidiagonals: A quotient belonging to a generalization of Euler's theorem.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 2, 0, 4, 6, 8, 3, 0, 5, 10, 20, 18, 6, 0, 6, 15, 40, 60, 48, 8, 0, 7, 21, 70, 150, 204, 108, 18, 0, 8, 28, 112, 315, 624, 640, 312, 30, 0, 9, 36, 168, 588, 1554, 2500, 2340, 810, 56, 0, 10, 45, 240, 1008, 3360, 7560, 11160, 8160, 2184, 96, 0

Offset: 1

Franz Vrabec, Aug 26 2021

The quotient T(n,k) = (k^n - k^(n-phi(n)))/n results from the generalization k^n == k^(n-phi(n)) (mod n) of Euler's theorem (see Sierpiński, p. 243).

The n-th row of the table is equal to the n-th row of A074650 iff n = p^j (p prime, j>=1).

			T(4,3) = (3^4 - 3^2)/4 = 18.
Square array starts:
  0, 1,   2,   3,    4,    5, ...
  0, 1,   3,   6,   10,   15, ...
  0, 2,   8,  20,   40,   70, ...
  0, 3,  18,  60,  150,  315, ...
  0, 6,  48, 204,  624, 1554, ...
  0, 8, 108, 640, 2500, 7560, ...

W. Sierpiński, Elementary Theory of Numbers, Warszawa, 1964.

Cf. A074650.

with(numtheory):
T:= proc(n, k) (k^n-k^(n-phi(n)))/n end:
seq(seq(T(i, 1+d-i), i=1..d), d=1..11);

T(n,k) = (k^n - k^(n - eulerphi(n)))/n; \\ Jinyuan Wang, Aug 28 2021

T(n,k) = (k^n - k^(n - phi(n)))/n.

More terms from Jinyuan Wang, Aug 28 2021

A383042 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) is the n-th term of the inverse Euler transform of j-> k^(j-1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 3, 0, 1, 4, 12, 15, 6, 0, 1, 5, 20, 42, 42, 9, 0, 1, 6, 30, 90, 156, 107, 18, 0, 1, 7, 42, 165, 420, 554, 294, 30, 0, 1, 8, 56, 273, 930, 1910, 2028, 780, 56, 0, 1, 9, 72, 420, 1806, 5155, 8820, 7350, 2128, 99, 0

Offset: 1

Seiichi Manyama, Apr 13 2025

			Square array begins:
  1,  1,   1,    1,    1,     1,     1, ...
  0,  1,   2,    3,    4,     5,     6, ...
  0,  2,   6,   12,   20,    30,    42, ...
  0,  3,  15,   42,   90,   165,   273, ...
  0,  6,  42,  156,  420,   930,  1806, ...
  0,  9, 107,  554, 1910,  5155, 11809, ...
  0, 18, 294, 2028, 8820, 28830, 77658, ...
  ...

Christian G. Bower, PARI programs for transforms, 2007.
N. J. A. Sloane, Maple programs for transforms, 2001-2020.

Columns k=1..5 give A000007, A059966, A065178, A065179, A065180.
Main diagonal gives A306173.
Cf. A065177 (another version).
Cf. A008683, A074650, A383033.

a(n, k) = sumdiv(n, d, moebius(n/d)*(k^d-(k-1)^d))/n;

A(n,k) = (1/n) * Sum_{d|n} mu(n/d) * (k^d - (k-1)^d).

A(n,k) = (1/n) * (k^n - (k-1)^n - Sum_{d

A(n,k) = A074650(n,k) - A074650(n,k-1).

Product_{n>=1} 1/(1 - x^n)^A(n,k) = (1 - (k-1)*x)/(1 - k*x).

G.f. of column k: Sum_{j>=1} mu(j) * log(1 + x^j/(1 - k*x^j)) / j.

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A006180 Witt vector *5!/5!.

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A363916 Array read by descending antidiagonals. A(n, k) = Sum_{d=0..k} A363914(k, d) * n^d.

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A383023 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) is the n-th term of the inverse Weigh transform of j-> k^j.

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A347277 Table T(n,k) read by downward antidiagonals: A quotient belonging to a generalization of Euler's theorem.

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A383042 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) is the n-th term of the inverse Euler transform of j-> k^(j-1).

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