A051170 -id:A051170 - cp's OEIS frontend

A245558 Square array read by antidiagonals: T(n,k) = number of n-tuples of nonnegative integers (u_0,...,u_{n-1}) satisfying Sum_{j=0..n-1} j*u_j == 1 mod n and Sum_{j=0..n-1} u_j = m.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 5, 5, 3, 1, 1, 3, 7, 8, 7, 3, 1, 1, 4, 9, 14, 14, 9, 4, 1, 1, 4, 12, 20, 25, 20, 12, 4, 1, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1, 5, 18, 40, 66, 75, 66, 40, 18, 5, 1, 1, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 1

Offset: 1

N. J. A. Sloane, Aug 07 2014

The array is symmetric; for the entries on or below the diagonal see A245559.
If the congruence in the definition is changed from Sum_{j=0..n-1} j*u_j == 1 mod n  to Sum_{j=0..n-1} j*u_j == 0 mod n we get the array shown in A241926, A047996, and A037306.
Differs from A011847 from row n = 9, k = 4 on; if the rows are surrounded by 0's, this yields A051168 without its rows 0 and 1, i.e., a(1) is A051168(2,1). - M. F. Hasler, Sep 29 2018
This array was first studied by Fredman (1975). - Petros Hadjicostas, Jul 10 2019

			Square array begins:
  1, 1,  1,  1,   1,   1,    1,    1,    1,    1, ...
  1, 1,  2,  2,   3,   3,    4,    4,    5,    5, ...
  1, 2,  3,  5,   7,   9,   12,   15,   18,   22, ...
  1, 2,  5,  8,  14,  20,   30,   40,   55,   70, ...
  1, 3,  7, 14,  25,  42,   66,   99,  143,  200, ...
  1, 3,  9, 20,  42,  75,  132,  212,  333,  497, ...
  1, 4, 12, 30,  66, 132,  245,  429,  715, 1144, ...
  1, 4, 15, 40,  99, 212,  429,  800, 1430, 2424, ...
  1, 5, 18, 55, 143, 333,  715, 1430, 2700, 4862, ...
  1, 5, 22, 70, 200, 497, 1144, 2424, 4862, 9225, ...
  ...
Reading by antidiagonals, we get:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  2,  1;
  1, 2,  3,  2,  1;
  1, 3,  5,  5,  3,   1;
  1, 3,  7,  8,  7,   3,   1;
  1, 4,  9, 14, 14,   9,   4,  1;
  1, 4, 12, 20, 25,  20,  12,  4,  1;
  1, 5, 15, 30, 42,  42,  30, 15,  5,  1;
  1, 5, 18, 40, 66,  75,  66, 40, 18,  5, 1;
  1, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 1;
  ...

Taylor Brysiewicz, Necklaces count polynomial parametric osculants, arXiv:1807.03408 [math.AG], 2018.
A. Elashvili, M. Jibladze, Hermite reciprocity for the regular representations of cyclic groups, Indag. Math. (N.S.) 9 (1998), no. 2, 233-238. MR1691428 (2000c:13006).
A. Elashvili, M. Jibladze, D. Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Algebraic Combin. 10 (1999), no. 2, 173-188. MR1719140 (2000j:05009). See p. 174.
M. L. Fredman, A symmetry relationship for a class of partitions, J. Combinatorial Theory Ser. A 18 (1975), 199-202.
I. M. Gessel and C. Reutenauer, Counting permutations with given cycle structure and descent set, J. Combin. Theory, Ser. A, 64, 1993, 189-215, Theorem 9.4.
J. E. Iglesias, Enumeration of closest-packings by the space group: a simple approach, Z. Krist. 221 (2006) 237-245, eq. (5).

This array is very similar to but different from A011847.
Cf. A051168, A092964, A241926, A047996, A037306, A245559.
Rows include A001840, A006918, A051170, A011796, A011797, A031164. Main diagonal is A022553.

# To produce the first 10 rows and columns (as on page 174 of the Elashvili et al. 1999 reference):
with(numtheory):
cnk:=(n,k) -> add(mobius(n/d)*d, d in divisors(gcd(n,k)));
anmk:=(n,m,k)->(1/(n+m))*add( cnk(d,k)*binomial((n+m)/d,n/d), d in divisors(gcd(n,m))); # anmk(n,m,k) is the value of a_k(n,m) as in Theorem 1, Equation (4), of the Elashvili et al. 1999 reference.
r2:=(n,k)->[seq(anmk(n,m,k),m=1..10)];
for n from 1 to 10 do lprint(r2(n,1)); od:

rows = 12;
cnk[n_, k_] := Sum[MoebiusMu[n/d] d, {d , Divisors[GCD[n, k]]}];
anmk[n_, m_, k_] := (1/(n+m)) Sum[cnk[d, k] Binomial[(n+m)/d, n/d], {d, Divisors[GCD[n, m]]}];
r2[n_, k_] := Table[anmk[n, m, k], {m, 1, rows}];
T = Table[r2[n, 1], {n, 1, rows}];
Table[T[[n-k+1, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 05 2018, from Maple *)

A011795 a(n) = floor(C(n,4)/5).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 3, 7, 14, 25, 42, 66, 99, 143, 200, 273, 364, 476, 612, 775, 969, 1197, 1463, 1771, 2125, 2530, 2990, 3510, 4095, 4750, 5481, 6293, 7192, 8184, 9275, 10472, 11781, 13209, 14763, 16450, 18278, 20254, 22386, 24682, 27150, 29799, 32637, 35673, 38916, 42375, 46060, 49980, 54145, 58565, 63250, 68211, 73458, 79002, 84854, 91025, 97527, 104371

Offset: 0

N. J. A. Sloane, David Broadhurst

a(n-1) = number of aperiodic necklaces (Lyndon words) with 5 black beads and n-5 white beads.

J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 147.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
David J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.
David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See pp. 3, 11, 19.
Index entries for sequences related to Lyndon words
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1,1,-4,6,-4,1).

Cf. A000031, A001037, A011915, A051168.
Same as A051170(n+1).
A column of triangle A011847.

[Floor(Binomial(n+1,5)/(n+1)): n in [0..70]]; // Vincenzo Librandi Jun 19 2012

seq(floor(binomial(n,4)/5), n=0.. 70); # Zerinvary Lajos, Jan 12 2009

CoefficientList[Series[x^5(1+x^3)/((1-x)^3(1-x^2)(1-x^5)),{x,0,70}],x] (* Vincenzo Librandi, Jun 19 2012 *)
CoefficientList[Series[x^4/5 (1/(1-x)^5-1/(1- x^5)),{x,0,70}],x] (* Herbert Kociemba, Oct 16 2016 *)

a(n)=binomial(n,4)\5 \\ Charles R Greathouse IV, Oct 07 2015

[binomial(n,4)//5 for n in range(71)] # G. C. Greubel, Oct 20 2024

G.f.: x^5*(1+x^3)/((1-x)^3*(1-x^2)*(1-x^5)) = x^5*(1-x+x^2)/((1-x)^5*(1+x+x^2+x^3+x^4)).

a(n) = floor(binomial(n+1,5)/(n+1)). - Gary Detlefs, Nov 23 2011

A050190 T(n,5), array T as in A050186; a count of aperiodic binary words.

Original entry on oeis.org

0, 6, 21, 56, 126, 250, 462, 792, 1287, 2002, 3000, 4368, 6188, 8568, 11628, 15500, 20349, 26334, 33649, 42504, 53125, 65780, 80730, 98280, 118755, 142500, 169911, 201376, 237336, 278256, 324625, 376992, 435897, 501942

Offset: 5

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 5..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1,2,-8,12,-8,2,-1,4,-6,4, -1).

Cf. A000389, A050186.

[n*floor(Binomial(n-1, 4)/5): n in [5..40]]; // G. C. Greubel, Nov 25 2017

Table[n*Floor[Binomial[n - 1, 4]/5], {n, 5, 50}] (* G. C. Greubel, Nov 25 2017 *)
Drop[CoefficientList[Series[(x^5*(3 + x^2 + x^3)*(2 - x + 2*x^2 + x^3 + x^4))/((1 - x)^4*(1 - x^5)^2), {x, 0, 50}], x], 4] (* G. C. Greubel, Nov 27 2017 *)

for(n=5,40, print1(n*floor(binomial(n-1, 4)/5), ", ")) \\ G. C. Greubel, Nov 25 2017

x='x+O('x^30); concat([0], Vec((x^5*(3 + x^2 + x^3)*(2 - x + 2*x^2 + x^3 + x^4))/((1 - x)^4*(1 - x^5)^2))) \\ G. C. Greubel, Nov 27 2017

a(n) = n * A051170(n).
From Ralf Stephan, Aug 18 2004: (Start)
G.f.: (x^5*(3 + x^2 + x^3)*(2 - x + 2*x^2 + x^3 + x^4))/((1 - x)^4*(1 - x^5)^2). (corrected by G. C. Greubel, Nov 27 2017)
a(n) = A000389(n) - [5 divides n]*n/5.
a(n) = n*floor(C(n-1, 4)/5). (End)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 2*a(n-5) - 8*a(n-6) + 12*a(n-7) - 8*a(n-8) + 2*a(n-9) - a(n-10) + 4*a(n-11) - 6*a(n-12) + 4*a(n-13) - a(n-14). - R. J. Mathar, May 20 2013

More terms from Ralf Stephan, Aug 18 2004

cp's OEIS Frontend

A245558 Square array read by antidiagonals: T(n,k) = number of n-tuples of nonnegative integers (u_0,...,u_{n-1}) satisfying Sum_{j=0..n-1} j*u_j == 1 mod n and Sum_{j=0..n-1} u_j = m.

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A011795 a(n) = floor(C(n,4)/5).

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A050190 T(n,5), array T as in A050186; a count of aperiodic binary words.

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