A053727 -id:A053727 - cp's OEIS frontend

A050186 Triangular array T read by rows: T(h,k) = number of binary words of k 1's and h-k 0's which are not a juxtaposition of 2 or more identical subwords.

1, 1, 1, 0, 2, 0, 0, 3, 3, 0, 0, 4, 4, 4, 0, 0, 5, 10, 10, 5, 0, 0, 6, 12, 18, 12, 6, 0, 0, 7, 21, 35, 35, 21, 7, 0, 0, 8, 24, 56, 64, 56, 24, 8, 0, 0, 9, 36, 81, 126, 126, 81, 36, 9, 0, 0, 10, 40, 120, 200, 250, 200, 120, 40, 10, 0, 0, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11

Offset: 0

Clark Kimberling

			For example, T(4,2) counts 1100,1001,0011,0110; T(2,1) counts 10, 01 (hence also counts 1010, 0101).
Rows:
  1;
  1,  1;
  0,  2,  0;
  0,  3,  3,  0;
  0,  4,  4,  4,  0;
  0,  5, 10, 10,  5,  0;

M. F. Hasler, A050186, rows 0..50, Sep 27 2018
M. F. Hasler, A050186, rows 0..100, Sep 27 2018
M. F. Hasler, A050186, rows 0..200, Sep 27 2018
N. J. A. Sloane, Transforms
Index entries for triangles and arrays related to Pascal's triangle

Same triangle as A053727 except this one includes column 0.

T(2n, n), T(2n+1, n) match A007727, A001700, respectively. Row sums match A027375.

T[n_, k_] := If[n == 0, 1, DivisorSum[GCD[k, n], MoebiusMu[#] Binomial[n/#, k/#]&]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 16 2022 *)

A050186(n,k)=sumdiv(gcd(n+!n,k),d,moebius(d)*binomial(n/d,k/d)) \\ M. F. Hasler, Sep 27 2018

MOEBIUS transform of A007318 Pascal's Triangle.

If rows n > 1 are divided by n, this yields the triangle A051168, which equals A245558 surrounded by 0's (except for initial terms). This differs from A011847 from row n = 9 on. - M. F. Hasler, Sep 29 2018

A165920 Number of 2-elements orbits of S3 action on irreducible polynomials of degree 3n, n > 0, over GF(2).

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 6, 10, 19, 33, 62, 112, 210, 387, 728, 1360, 2570, 4845, 9198, 17459, 33288, 63519, 121574, 232960, 447392, 860265, 1657009, 3195465, 6170930, 11930100, 23091222, 44738560, 86767016, 168428805, 327235602, 636289024, 1238188770, 2411205111, 4698767640, 9162588158, 17878237850

Offset: 1

Jean Francis Michon, Philippe Ravache (philippe.ravache(AT)univ-rouen.fr), Sep 30 2009

Arndt's PARI code computes a(n) as the sum, divided by n, of every 3rd term in row n of L = A050186 = Möbius transform of binomials, starting with k = (1-n) mod 3 (nonnegative remainder), where k = 0 and k = n give L(n, k) = 0 and can be omitted. Cf. A053727, EXAMPLE and second PROGRAM. - M. F. Hasler, Sep 27 2018

			Illustrating computation via L = A050186, cf. COMMENTS: a(1) = [L(1,0)] = 0. a(2) = [L(2,2)] = 0. a(3) = L(3,1)/3 = 3/3 = 1. a(4) = ([L(4,0)] + L(4,3))/4 = 4/4 = 1. a(5) = (L(5,2) + [L(5,5)])/5 = 10/5 = 2. In [...] are terms L(n,0) = L(n,n) = 0.

Robert Israel, Table of n, a(n) for n = 1..3331
J. E. Iglesias, Enumeration of polytypes of MX and MX_2 through the use of the symmetry of the Zhadanov symbol, Acta Cryst. A 62 (3) (2006) 176-194, Table 1.
J.-F. Michon, P. Ravache, On different families of invariant irreducible polynomials over F_2, Finite fields & Applications 16 (2010) 163-174.

This sequence is the half of A165912 (the number of alternate polynomials). A001037 is the enumeration by degree of the polynomials of I. A000048 is the number of 3-elements orbits of S3 action on I.

f:= proc(n) local D,d;
  D:=remove(d -> (n/3/d)::integer, numtheory:-divisors(n));
  add(numtheory:-mobius(n/d)*(2^d - (-1)^d),d=D)/(3*n)
end proc:
map(f, [$1..100]); # Robert Israel, Jul 14 2019

a[n_] := Sum[If[Mod[n/d, 3] == 0, 0, MoebiusMu[n/d]*(2^d - (-1)^d)/(3n)], {d, Divisors[n]}];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 02 2023 *)

L(n, k) = sumdiv(gcd(n,k), d, moebius(d) * binomial(n/d, k/d) );
a(n) = sum(k=0, n, if( (n+k)%3==1, L(n, k), 0 ) ) / n;
vector(55,n,a(n))
/* Joerg Arndt, Jun 28 2012 */

A165920(n,k=(1-n)%3)=sum(i=0,(n-k)\3,A050186(n,k+3*i))\n \\ For illustration. - M. F. Hasler, Sep 30 2018

a(n) = (sum_{d|n, n/d != 0 mod 3} mu(n/d)*(2^d - (-1)^d))/(3n).

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A050186 Triangular array T read by rows: T(h,k) = number of binary words of k 1's and h-k 0's which are not a juxtaposition of 2 or more identical subwords.

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A165920 Number of 2-elements orbits of S3 action on irreducible polynomials of degree 3n, n > 0, over GF(2).

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