A108299 -id:A108299 - cp's OEIS frontend

A078031 Expansion of (1-x)/(1 + x^2 - x^3).

1, -1, -1, 2, 0, -3, 2, 3, -5, -1, 8, -4, -9, 12, 5, -21, 7, 26, -28, -19, 54, -9, -73, 63, 64, -136, -1, 200, -135, -201, 335, 66, -536, 269, 602, -805, -333, 1407, -472, -1740, 1879, 1268, -3619, 611, 4887, -4230, -4276, 9117, 46, -13393, 9071, 13439, -22464, -4368, 35903, -18096

Offset: 0

N. J. A. Sloane, Nov 17 2002

The Ca2 sums, see A180662, of triangle A108299 equal the terms of this sequence. - Johannes W. Meijer, Aug 14 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,-1,1).

Cf. A077961, A108299, A180662. - Johannes W. Meijer, Aug 14 2011

a:=[1,-1,-1];; for n in [4..60] do a[n]:=-a[n-2]+a[n-3]; od; a; # G. C. Greubel, Aug 05 2019

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1-x)/(1+x^2-x^3) )); // G. C. Greubel, Aug 05 2019

A078031 := proc(n) option remember: coeftayl((1-x)/(1+x^2-x^3),x=0,n) end: seq(A078031(n), n=0..60); # Johannes W. Meijer, Aug 14 2011

CoefficientList[Series[(1-x)/(1+x^2-x^3),{x,0,60}],x] (* or *) LinearRecurrence[{0,-1,1},{1,-1,-1},60] (* Harvey P. Dale, Apr 08 2012 *)

Vec((1-x)/(1+x^2-x^3)+O(x^60)) \\ Charles R Greathouse IV, Sep 26 2012

((1-x)/(1+x^2-x^3)).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Aug 05 2019

G.f.: (1-x)/(1 + x^2 - x^3).

a(n) = -a(n-2) + a(n-3); a(0)=1, a(1)=-1, a(2)=-1. - Harvey P. Dale, Apr 08 2012

A134668 Period 6: repeat [1, -1, 0, 0, -1, 1].

Original entry on oeis.org

1, -1, 0, 0, -1, 1, 1, -1, 0, 0, -1, 1, 1, -1, 0, 0, -1, 1, 1, -1, 0, 0, -1, 1, 1, -1, 0, 0, -1, 1, 1, -1, 0, 0, -1, 1, 1, -1, 0, 0, -1, 1, 1, -1, 0, 0, -1, 1, 1, -1, 0, 0, -1, 1, 1, -1, 0, 0, -1, 1, 1, -1, 0, 0, -1, 1, 1, -1, 0, 0, -1, 1, 1, -1, 0, 0, -1, 1

Offset: 0

Paul Curtz, Jan 26 2008

The Fi2 sums, see A180662, of triangle A108299 equal the terms of this sequence. - Johannes W. Meijer, Aug 11 2011

			G.f. = 1 - x - x^4 + x^5 + x^6 - x^7 - x^10 + x^11 + x^12 - x^13 - x^16 + ...

Index entries for linear recurrences with constant coefficients, signature (0,-1,0,-1).

Cf. A108299, A134667, A180662.

&cat [[1, -1, 0, 0, -1, 1]^^20]; // Wesley Ivan Hurt, Jun 20 2016

A134668 :=proc(n): (1/6)*(-2*((n+1) mod 6)+((n+2) mod 6)-((n+4) mod 6)+2*((n+5) mod 6)) end: seq(A134668(n), n=0..74); # Johannes W. Meijer, Aug 14 2011

PadRight[{},120,{1,-1,0,0,-1,1}] (* or *) LinearRecurrence[{0,-1,0,-1},{1,-1,0,0},120] (* Harvey P. Dale, Dec 03 2012 *)

{a(n)=[1, -1, 0, 0, -1, 1][n%6+1]}; /* Michael Somos, Feb 08 2008 */

First differences of A134667.
Euler transform of length 6 sequence [-1, 0, 0, -1, 0, 1]. - Michael Somos, Feb 08 2008
a(n) = a(-1-n) for all n in Z. - Michael Somos, Feb 08 2008
G.f.: (1-x)*(1-x^4) / (1-x^6) = (1-x)*(1+x^2) / ((1-x+x^2)*(1+x+x^2)) = (1-x+x^2-x^3) / (1+x^2+x^4).
a(6*n + 2) = a(6*n + 3) = 0. - Michael Somos, Oct 16 2015
From Wesley Ivan Hurt, Jun 20 2016: (Start)
a(n) + a(n-2) + a(n-4) = 0 for n>3.
a(n) = cos(n*Pi/6) * (3*cos(n*Pi/2) + 2*sqrt(3)*sin(n*Pi/6) - 3*sqrt(3)*sin(n*Pi/2))/3. (End)

A191315 Sum of the heights of all dispersed Dyck paths of length n (i.e., of Motzkin paths of length n with no (1,0) steps at positive heights).

Original entry on oeis.org

0, 0, 1, 2, 6, 11, 27, 50, 115, 216, 481, 913, 1992, 3809, 8192, 15748, 33512, 64685, 136546, 264422, 554686, 1077055, 2248105, 4375221, 9095238, 17735812, 36745504, 71776633, 148288346, 290092160, 597876033, 1171153370, 2408702852, 4723840544, 9697826974, 19038878297

Offset: 0

Emeric Deutsch, May 31 2011

nonn

a(n) = Sum_{k>=0} k * A191314(n,k).

			a(4)=6 because the sum of the heights of the paths HHHH, HHUD, HUDH, UDHH, UDUD, and UUDD is 0+1+1+1+1+2=6; here U=(1,1), H=(1,0), D=(1,-1).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A108299, A191314.

F[0] := 1: F[1] := 1-z: for k from 2 to 36 do F[k] := sort(expand(F[k-1]-z^2*F[k-2])) end do: G := sum(j*z^(2*j)/(F[j]*F[j+1]), j = 0 .. 34): Gser := series(G, z = 0, 40): seq(coeff(Gser, z, n), n = 0 .. 35);
# second Maple program:
b:= proc(x, y, m) option remember;
      `if`(y>x or y<0, 0, `if`(x=0, m, b(x-1, y-1, m)+
      `if`(y=0, b(x-1, y, m), 0)+b(x-1, y+1, max(m, y+1))))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 13 2017

b[x_, y_, m_] := b[x, y, m] = If[y > x || y < 0, 0, If[x == 0, m, b[x - 1, y - 1, m] + If[y == 0, b[x - 1, y, m], 0] + b[x - 1, y + 1, Max[m, y + 1]]]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 16 2017, after Alois P. Heinz *)

G.f.: G(z) = Sum_{j>=0}(jz^(2j)/(F(j)F(j+1))), where F(k) are polynomials in z defined by F(0)=1, F(1)=1-z, F(k)=F(k-1)-z^2*F(k-2) for k>=2. The coefficients of these polynomials form the triangle A108299.

A193669 Expansion of o.g.f.(1-x^4)/(1-x+x^8).

Original entry on oeis.org

1, 1, 1, 1, 0, 0, 0, 0, -1, -2, -3, -4, -4, -4, -4, -4, -3, -1, 2, 6, 10, 14, 18, 22, 25, 26, 24, 18, 8, -6, -24, -46, -71, -97, -121, -139, -147, -141, -117, -71, 0, 97, 218, 357, 504, 645, 762, 833, 833, 736, 518, 161, -343, -988, -1750, -2583, -3416, -4152

Offset: 0

Johannes W. Meijer, Aug 11 2011

The Gi1 sums, see A180662, of triangle A108299 equal the terms of this sequence.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,-1).

Cf. A108299, A180662.

A193669 := proc(n) option remember: coeftayl((1-x^4) / (1-x+x^8) ,x=0,n) end: seq(A193669(n), n=0..57);

CoefficientList[Series[(1-x^4)/(1-x+x^8),{x,0,80}],x] (* or *) LinearRecurrence[ {1,0,0,0,0,0,0,-1},{1,1,1,1,0,0,0,0},80] (* Harvey P. Dale, Jul 16 2014 *)

G.f.: -(x-1)*(1+x)*(x^2+1) / ( (x^2-x+1)*(x^6+x^5-x^3-x^2+1) ).

a(n) = a(n-1) - a(n-8), a(0) = a(1) = a(2) = a(3) = 1, a(4) = a(5) = a(6) = a(7) = 0.

A193884 Expansion of o.g.f. (1-x^2)/(1-x+x^4).

Original entry on oeis.org

1, 1, 0, 0, -1, -2, -2, -2, -1, 1, 3, 5, 6, 5, 2, -3, -9, -14, -16, -13, -4, 10, 26, 39, 43, 33, 7, -32, -75, -108, -115, -83, -8, 100, 215, 298, 306, 206, -9, -307, -613, -819, -810, -503, 110, 929, 1739, 2242, 2132, 1203, -536, -2778, -4910, -6113, -5577

Offset: 0

Johannes W. Meijer, Aug 11 2011

The Kn11 sums, see A180662, of triangle A108299 equal the terms of this sequence.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,-1).

Cf. A099530, A108299, A180662.

A193884 := proc(n) option remember: if n=0 then 1 elif n=1 then 1 elif n=2 then 0 elif n=3 then 0 elif n>=4 then procname(n-1)-procname(n-4) fi: end: seq(A193884(n), n=0..54);

CoefficientList[Series[(1-x^2)/(1-x+x^4),{x,0,80}],x] (* or *) LinearRecurrence[{1,0,0,-1},{1,1,0,0},80] (* Harvey P. Dale, Jul 15 2020 *)

G.f.: (1+x)*(1-x)/(1-x+x^4).
a(n) = a(n-1)-a(n-4), a(0) = a(1) = 1, a(2) = a(3) = 0.
a(n) = A099530(n) - A099530(n-2).

A193885 a(n) = 3a(n-1) - 3a(n-2) + a(n-3) - a(n-4), n>=4; a(0) = 1, a(1) = 1, a(2) = 2, a(3) = 3.

Original entry on oeis.org

1, 1, 2, 3, 3, 1, -5, -18, -41, -75, -115, -143, -118, 35, 431, 1213, 2499, 4254, 6047, 6665, 3609, -7375, -32334, -77933, -147781, -234503, -305765, -283634, -20329, 718653, 2239077, 4824577, 8495482, 12533139, 14698471, 10166901, -9557053, -57006530

Offset: 0

Johannes W. Meijer, Aug 11 2011

The Ze1 sums, see A180662, of triangle A108299 equal the terms of this sequence.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,-1).

Cf. A099531, A108299, A180662.

I:=[1, 1, 2, 3 ]; [n le 4 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 10 2012

A193885 := proc(n) option remember: if n=0 then 1 elif n=1 then 1 elif n=2 then 2 elif n=3 then 3 elif n>=4 then 3*procname(n-1)-3*procname(n-2)+procname(n-3)-procname(n-4) fi: end: seq(A193885(n),n=0..37);

CoefficientList[Series[(1-x)*(1-x+x^2)/(1-3*x+3*x^2-x^3+x^4),{x,0,50}],x] (* Vincenzo Librandi, Jul 10 2012 *)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) - a(n-4), n>=4; a(0) = 1, a(1) = 1, a(2) = 2, a(3) = 3.
G.f.: (1-x)*(1-x+x^2)/(1-3*x+3*x^2-x^3+x^4).
a(n) = (-1)^(n+1)*(A099531(n+4) + 2*A099531(n+3) + 2*A099531(n+2) + A099531(n+1)).

A209235 Rectangular array read by antidiagonals, with entry k in row n given by T(n,k) = 2^{k-1}Sum_{j=1..n} (cos((2j-1)Pi/(2n+1)))^{k-1}.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 1, 3, 1, 5, 1, 5, 4, 1, 6, 1, 7, 4, 7, 1, 8, 1, 11, 4, 19, 16, 18, 1, 9, 1, 13, 4, 25, 16, 38, 29, 1, 10, 1, 15, 4, 31, 16, 58, 57, 47, 1, 11, 1, 17, 4, 37, 16, 78, 64, 117, 76, 1, 12, 1, 19, 4, 43, 16, 98, 64, 187, 193, 123, 1

Offset: 1

L. Edson Jeffery, Jan 12 2013

Antidiagonal sums: {1,3,5,9,16,26,46,78,136,...}.

			Array begins as
.1..1...1..1...1...1
.2..1...3..4...7..11
.3..1...5..4..13..16
.4..1...7..4..19..16
.5..1...9..4..25..16
.6..1..11..4..31..16

Rows: cf. A000012, A000032, A094649, A189234, A216605, etc.

Cf. A185095, A186740.

T(n,k) = 2^{k-1}*Sum_{j=1..n} (cos((2*j-1)*Pi/(2*n+1)))^{k-1}.
Empirical g.f. for row n: F(x) = (Sum_{u=0..n-1} A122765(n,n-1-u)*x^u)/(Sum_{v=0..n} A108299(n,v)*x^v).
Empirical: odd column first differences tend to A000984 = {1, 2, 6, 20, 70, 252, ...} (central binomial coefficients).

cp's OEIS Frontend

A078031 Expansion of (1-x)/(1 + x^2 - x^3).

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A134668 Period 6: repeat [1, -1, 0, 0, -1, 1].

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A191315 Sum of the heights of all dispersed Dyck paths of length n (i.e., of Motzkin paths of length n with no (1,0) steps at positive heights).

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A193669 Expansion of o.g.f.(1-x^4)/(1-x+x^8).

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A193884 Expansion of o.g.f. (1-x^2)/(1-x+x^4).

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A193885 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) - a(n-4), n>=4; a(0) = 1, a(1) = 1, a(2) = 2, a(3) = 3.

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A209235 Rectangular array read by antidiagonals, with entry k in row n given by T(n,k) = 2^{k-1}*Sum_{j=1..n} (cos((2*j-1)*Pi/(2*n+1)))^{k-1}.

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A193885 a(n) = 3a(n-1) - 3a(n-2) + a(n-3) - a(n-4), n>=4; a(0) = 1, a(1) = 1, a(2) = 2, a(3) = 3.

A209235 Rectangular array read by antidiagonals, with entry k in row n given by T(n,k) = 2^{k-1}Sum_{j=1..n} (cos((2j-1)Pi/(2n+1)))^{k-1}.