A050186 -id:A050186 - cp's OEIS frontend

A050870 T(h,k) = binomial(h,k) - A050186(h,k).

0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 3, 2, 3, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 4, 0, 6, 0, 4, 0, 1, 1, 0, 0, 3, 0, 0, 3, 0, 0, 1, 1, 0, 5, 0, 10, 2, 10, 0, 5, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 6, 4, 15, 0, 24, 0, 15, 4, 6, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 7, 0, 21, 0, 35, 2, 35, 0, 21, 0, 7, 0, 1, 1, 0

Offset: 0

Clark Kimberling

T(h,k) = number of periodic binary words of k 1's and h-k 0's.

			0;
0,0;
1,0,1;
1,0,0,1;
1,0,2,0,1;
1,0,0,0,0,1;
1,0,3,2,3,0,1;
1,0,0,0,0,0,0,1;
1,0,4,0,6,0,4,0,1;
1,0,0,3,0,0,3,0,0,1;
1,0,5,0,10,2,10,0,5,0,1;

Cf. A007318. Different from A053200.

A050186 := proc(n,k)
        if n = 0 then
                1;
        else
        add (numtheory[mobius](d)*binomial(n/d,k/d),d =numtheory[divisors](igcd(n,k))) ;
        end if;
end proc:
A050870 := proc(n,k)
        binomial(n,k)-A050186(n,k) ;
end proc:
seq(seq(A050870(n,k),k=0..n),n=0..20) ; # R. J. Mathar, Sep 24 2011

T[n_, k_] := Binomial[n, k] - If[n == 0, 1, Sum[MoebiusMu[d] Binomial[n/d, k/d], {d, Divisors[GCD[n, k]]}]];
Table[T[n, k], {n, 0, 15}, {k, 0, n}] (* Jean-François Alcover, Jul 01 2019 *)

Edited by N. J. A. Sloane, Aug 29 2008

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

A050188 T(n,3), array T as in A050186; a count of aperiodic binary words.

Original entry on oeis.org

0, 4, 10, 18, 35, 56, 81, 120, 165, 216, 286, 364, 450, 560, 680, 810, 969, 1140, 1323, 1540, 1771, 2016, 2300, 2600, 2916, 3276, 3654, 4050, 4495, 4960, 5445, 5984, 6545, 7128, 7770, 8436, 9126, 9880, 10660, 11466, 12341, 13244, 14175

Offset: 3

Clark Kimberling

nonn

T(n, 3)=n*A001840(n).

A050189 T(n,4), array T as in A050186; a count of aperiodic binary words.

Original entry on oeis.org

0, 5, 12, 35, 64, 126, 200, 330, 480, 715, 980, 1365, 1792, 2380, 3024, 3876, 4800, 5985, 7260, 8855, 10560, 12650, 14872, 17550, 20384, 23751, 27300, 31465, 35840, 40920, 46240, 52360, 58752, 66045, 73644, 82251, 91200

Offset: 4

Clark Kimberling

nonn

Cf. A006918, A050186.

Seems to be n * A006918.
From Chai Wah Wu, Jun 11 2016: (Start)
a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8) for n > 11 (conjectured).
G.f.: x^5*(5 + 2*x + x^2)/((1 - x)^5*(1 + x)^3) (conjectured). (End)

A050191 T(n,6), array T as in A050186; a count of aperiodic binary words.

Original entry on oeis.org

0, 7, 24, 81, 200, 462, 900, 1716, 2968, 4995, 7952, 12376, 18468, 27132, 38640, 54243, 74448, 100947, 134352, 177100, 229944, 295974, 376376, 475020, 593280, 736281, 905632, 1107513, 1344224, 1623160, 1946916

Offset: 6

Clark Kimberling

nonn

Cf. A011796, A050186.

Seems to be n*A011796.

A051192 T(n,7), array T as in A050186; a count of aperiodic binary words.

Original entry on oeis.org

0, 8, 36, 120, 330, 792, 1716, 3430, 6435, 11440, 19448, 31824, 50388, 77520, 116277, 170544, 245157, 346104, 480700, 657800, 888030, 1184036, 1560780, 2035800, 2629575, 3365856, 4272048, 5379616, 6724515, 8347680, 10295472

Offset: 7

Clark Kimberling

nonn

Seems to be n*A011797.

A051195 T(2n+2,n), array T as in A050186; a count of aperiodic binary words.

Original entry on oeis.org

0, 4, 12, 56, 200, 792, 2968, 11440, 43632, 167960, 646184, 2496144, 9655984, 37442160, 145416240, 565722720, 2203937120, 8597496600, 33577908232, 131282408400, 513791254704, 2012616400080, 7890369761872, 30957699535776

Offset: 0

Clark Kimberling

nonn

C(2n+2, n)-[n even]*C(n+1, n/2). - Ralf Stephan, Mar 19 2004

Conjecture: -(n+2)*(n-1)*a(n) +2*(n+1)*(3*n-4)*a(n-1) +4*(-n^2+5*n-1)*a(n-2) -8*n*(3*n-4)*a(n-3) +16*(n-1)*(2*n-5)*a(n-4)=0. - R. J. Mathar, May 20 2013

A051196 T(2n+3,n), array T as in A050186; a count of aperiodic binary words.

Original entry on oeis.org

0, 5, 21, 81, 330, 1287, 4995, 19448, 75582, 293895, 1144066, 4457400, 17383734, 67863915, 265182525, 1037157858, 4059928950, 15905368710, 62359142274, 244662670200, 960566918220, 3773655743715, 14833897694226, 58343356817424

Offset: 0

Clark Kimberling

nonn

A051197 T(2n+4,n), array T as in A050186; a count of aperiodic binary words.

Original entry on oeis.org

0, 6, 24, 120, 480, 2002, 7952, 31824, 125760, 497420, 1960464, 7726160, 30418752, 119759850, 471424160, 1855967520, 7307828352, 28781143380, 113380093840, 446775310800, 1761038703424, 6943526580276, 27385654785504

Offset: 0

Clark Kimberling

nonn

A051198 T(2n+5,n), array T as in A050186; a count of aperiodic binary words.

Original entry on oeis.org

0, 7, 36, 165, 715, 3000, 12376, 50388, 203490, 817190, 3268750, 13037895, 51895935, 206253075, 818809200, 3247943125, 12875774670, 51021117810, 202112640600, 800472431850, 3169870830000, 12551759587422, 49699896548176

Offset: 0

Clark Kimberling

nonn

cp's OEIS Frontend

A050870 T(h,k) = binomial(h,k) - A050186(h,k).

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

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A050188 T(n,3), array T as in A050186; a count of aperiodic binary words.

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A050189 T(n,4), array T as in A050186; a count of aperiodic binary words.

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A050191 T(n,6), array T as in A050186; a count of aperiodic binary words.

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A051192 T(n,7), array T as in A050186; a count of aperiodic binary words.

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A051195 T(2n+2,n), array T as in A050186; a count of aperiodic binary words.

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A051196 T(2n+3,n), array T as in A050186; a count of aperiodic binary words.

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A051197 T(2n+4,n), array T as in A050186; a count of aperiodic binary words.

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A051198 T(2n+5,n), array T as in A050186; a count of aperiodic binary words.

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