A057083 -id:A057083 - cp's OEIS frontend

A115052 Expansion of 1/(3x^2 - 3x + 1)^2.

1, 6, 21, 54, 108, 162, 135, -162, -1053, -2916, -5832, -8748, -8019, 4374, 41553, 118098, 236196, 354294, 334611, -118098, -1476225, -4251528, -8503056, -12754584, -12223143, 3188646, 49424013, 143489070, 286978140, 430467210, 416118303, -86093442, -1592728677

Offset: 0

Roger L. Bagula, Feb 28 2006

q=1 coefficient expansion of hierarchical lattice renormalization polynomial.

Heinz-Otto Peitgen and Peter Richter (editors), The Beauty of Fractals, Springer-Verlag, New York, 1986, p. 146.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (6,-15,18,-9).

Autoconvolution of A057083.

I:=[1,6,21,54]; [n le 4 select I[n] else 6*Self(n-1)-15*Self(n-2)+18*Self(n-3)-9*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Sep 20 2011

A115052 := proc(n) 1/(3*x^2-3*x+1)^2 ; coeftayl(%,x=0,n) ; end proc: # R. J. Mathar, Sep 17 2011

From Stefano Spezia, Sep 01 2025: (Start)
a(n) = 3^(n/2)*((1 - n)*cos(n*Pi/6) + sqrt(3)*(3 + n)*sin(n*Pi/6)).
E.g.f.: exp(3*x/2)*(cos(sqrt(3)*x/2) + sqrt(3)*(3 + 2*x)*sin(sqrt(3)*x/2)). (End)

A134581 a(n) = 4a(n-1) - 7a(n-2) + 6a(n-3) - 3a(n-4), starting with 0, 1, 2, 3.

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 0, -13, -40, -81, -122, -122, 0, 365, 1094, 2187, 3280, 3280, 0, -9841, -29524, -59049, -88574, -88574, 0, 265721, 797162, 1594323, 2391484, 2391484, 0, -7174453, -21523360, -43046721, -64570082, -64570082, 0

Offset: 0

Paul Curtz, Jan 23 2008

Index entries for linear recurrences with constant coefficients, signature (4,-7,6,-3).

LinearRecurrence[{4, -7, 6, -3}, {0, 1, 2, 3}, 50] (* Harvey P. Dale, Dec 06 2013 *)
a[ n_] := Nest[# + RotateRight @ #&, {0, -1, 0, 0, 0, 1}, n][[1]]; (* Michael Somos, Jan 18 2023 *)

G.f.: x*(1-2*x+2*x^2)/((1-x+x^2)*(1-3*x+3*x^2)). - Jaume Oliver Lafont, Aug 30 2009
a(n) = A140343(n+3) - 2*A140343(n+2) + 2*A140343(n+1). - R. J. Mathar, Nov 21 2012
From Peter Bala, Jul 24 2017: (Start)
a(6*n) = 0;
a(6*n+1) = ((-1)^n*3^(3*n) + 1)/2;
a(6*n+2) = ((-1)^n*3^(3*n+1) + 1)/2;
a(6*n+3) = (-1)^n*3^(3*n+1);
a(6*n+4) = a(6*n+5) = ((-1)^n*3^(3*n+2) - 1)/2.
The o.g.f. A(x) satisfies (1 - x)*A(x) = x*A(1 - x).
Logarithmic g.f.: (1/sqrt(3))*arctan(sqrt(3)*x*(1 - x)/(1 - 2*x)) = Sum_{n >= 1} a(n)*x^n/n.
Sum_{n >= 1} a(n)/(n*2^n) = Pi/(2*sqrt(3)). (End)
a(n) = (3^(n/2) * sin(Pi*n/6) + sin(Pi*n/3)) / sqrt(3). - Peter Luschny, Jul 24 2017
2*a(n) = A010892(n-1) + A057083(n-1). - R. J. Mathar, Oct 03 2021
a(n) = -26*a(n-6) + 27*a(n-12) for all n in Z. - Michael Somos, Jan 18 2023

A167925 Triangle, T(n, k) = (sqrt(k+1))^(n-1)*ChebyshevU(n-1, sqrt(k+1)/2), read by rows.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 0, 2, 6, 12, -1, 0, 9, 32, 75, -1, -4, 9, 80, 275, 684, 0, -8, 0, 192, 1000, 3240, 8232, 1, -8, -27, 448, 3625, 15336, 47677, 122368, 1, 0, -81, 1024, 13125, 72576, 276115, 835584, 2158569, 0, 16, -162, 2304, 47500, 343440, 1599066, 5705728, 16953624, 44010000

Offset: 0

Roger L. Bagula, Nov 15 2009

			Triangle begins as:
   0;
   1,  1;
   1,  2,   3;
   0,  2,   6,   12;
  -1,  0,   9,   32,    75;
  -1, -4,   9,   80,   275,   684;
   0, -8,   0,  192,  1000,  3240,   8232;
   1, -8, -27,  448,  3625, 15336,  47677, 122368;
   1,  0, -81, 1024, 13125, 72576, 276115, 835584, 2158569;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A009545, A030191, A030192, A030240, A057083, A057084, A057085, A057086, A099087, A128834, A190871, A190873.

A167925:= func< n,k | Round((Sqrt(k+1))^(n-1)*Evaluate(ChebyshevSecond(n), Sqrt(k+1)/2)) >;
[A167925(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 11 2023

(* First program *)
m[k_]= {{k,1}, {-1,1}};
v[0, k_]:= {0,1};
v[n_, k_]:= v[n, k]= m[k].v[n-1,k];
T[n_, k_]:= v[n, k][[1]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten
(* Second program *)
A167925[n_, k_]:= (Sqrt[k+1])^(n-1)*ChebyshevU[n-1, Sqrt[k+1]/2];
Table[A167925[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 11 2023 *)

def A167925(n,k): return (sqrt(k+1))^(n-1)*chebyshev_U(n-1, sqrt(k+1)/2)
flatten([[A167925(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 11 2023

T(n, k) = (-1)^(n+1) * [x^(n-1)]( 1/(1 + (k+1)*x + (k+1)*x^2) ). - Francesco Daddi, Aug 04 2011 (modified by G. C. Greubel, Sep 11 2023)
From G. C. Greubel, Sep 11 2023: (Start)
T(n, k) = (sqrt(k+1))^(n-1)*ChebyshevU(n-1, sqrt(k+1)/2).
T(n, 0) = A128834(n).
T(n, 1) = A009545(n) = A099087(n-1).
T(n, 2) = A057083(n-1).
T(n, 3) = A001787(n).
T(n, 4) = A030191(n-1).
T(n, 5) = A030192(n-1).
T(n, 6) = A030240(n-1).
T(n, 7) = A057084(n-1).
T(n, 8) = A057085(n).
T(n, 9) = A057086(n-1).
T(n, 10) = A190871(n).
T(n, 11) = A190873(n). (End)

Edited by G. C. Greubel, Sep 11 2023

A194084 Triangle read by rows: a(n)=A135929(n) + A192011(n). Row n gives coefficients of polynomials BC(n,x) in order of decreasing exponents.

Original entry on oeis.org

0, 3, 0, 3, 0, 3, 3, 0, 0, 0, 3, 0, -3, 0, -3, 3, 0, -6, 0, -3, 0, 3, 0, -9, 0, 0, 0, 3, 3, 0, -12, 0, 6, 0, 6, 0, 3, 0, -15, 0, 15, 0, 6, 0, -3, 3, 0, -18, 0, 27, 0, 0, 0, -9, 0

Offset: 0

Paul Curtz, Aug 14 2011

0,
3, 0,
3, 0,  3,
3, 0,  0, 0,
3, 0, -3, 0, -3,
3, 0, -6, 0, -3, 0.
Multiples of 3.
Row sum (from the second) is period 6: 3*A057079(n),"from" A057083 (scaled Chebyshev U(n,x)).
If a(0)=-3, a(n)=3*A192174(n).

			a(0)=1-1=0, a(1)=1+2=3, a(2)=0+0=0.

BC(0,x)=0, BC(1,x)=3*x, BC(2,x)=3*x^2+3, BC(n,x)=x*BC(n-1,x) - BC(n-2,x), n > 2.

A307395 Expansion of 1/((1 - x) * ((1 - x)^3 + x^3)).

Original entry on oeis.org

1, 4, 10, 19, 28, 28, 1, -80, -242, -485, -728, -728, 1, 2188, 6562, 13123, 19684, 19684, 1, -59048, -177146, -354293, -531440, -531440, 1, 1594324, 4782970, 9565939, 14348908, 14348908, 1, -43046720, -129140162, -258280325, -387420488, -387420488, 1, 1162261468

Offset: 0

Seiichi Manyama, Apr 07 2019

Seiichi Manyama, Table of n, a(n) for n = 0..4000
Index entries for linear recurrences with constant coefficients, signature (4,-6,3).

Column 5 of A307394.

Partial sums of A057083.

LinearRecurrence[{4, -6, 3}, {1, 4, 10}, 38] (* Amiram Eldar, May 13 2021 *)

{a(n) = sum(k=0, n\3, (-1)^k*binomial(n+3, 3*k+3))}

N=66; x='x+O('x^N); Vec(1/((1-x)*((1-x)^3+x^3)))

a(n) = Sum_{k=0..floor(n/3)} (-1)^k*binomial(n+3,3*k+3).
a(n) = 4*a(n-1) - 6*a(n-2) + 3*a(n-3) for n > 2.
a(6*n) = 1.
a(n) = 1 - A057681(n+3). - Yomna Bakr and Greg Dresden, Apr 22 2024

A099447 An Alexander sequence for the knot 6_3.

Original entry on oeis.org

1, 3, 4, 0, -13, -30, -29, 24, 140, 243, 130, -429, -1348, -1752, 67, 5346, 11795, 10608, -11180, -56541, -93694, -42525, 182452, 535440, 660179, -106782, -2197373, -4613112, -3832996, 5081235, 22766722, 36008115

Offset: 0

Paul Barry, Oct 16 2004

The denominator is a parameterization of the Alexander polynomial for the knot 6_3. 1/(1-3*x+5*x^2-3*x^3+x^4) is the image of the g.f. of A057083 under the modified Chebyshev transform A(x)->(1/(1+x^2)^2)A(x/(1+x^2)).

Dror Bar-Natan, The Rolfsen Knot Table
Index entries for linear recurrences with constant coefficients, signature (3,-5,3,-1).

LinearRecurrence[{3,-5,3,-1},{1,3,4,0,-13},40] (* Harvey P. Dale, Oct 07 2017 *)

G.f.: (1-x)*(1+x)*(1+x^2)/(1-3x+5x^2-3x^3+x^4); - corrected Nov 24 2012

a(n)=A099446(n)-A099446(n-2).

A201947 Triangle T(n,k), read by rows, given by (1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,-1,1,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 2, 2, 0, 3, 5, 1, -1, 5, 10, 4, -2, -1, 8, 20, 12, -4, -4, 0, 13, 38, 31, -4, -13, -2, 1, 21, 71, 73, 3, -33, -11, 3, 1, 34, 130, 162, 34, -74, -42, 6, 6, 0, 55, 235, 344, 128, -146, -130, 0, 24, 3, -1

Offset: 0

Philippe Deléham, Dec 06 2011

Row-reversed variant of A123585. Row sums: 2^n.

			Triangle begins:
1
1, 1
2, 2, 0
3, 5, 1, -1
5, 10, 4, -2, -1
8, 20, 12, -4, -4, 0
13, 38, 31, -4, -13, -2, 1
21, 71, 73, 3, -33, -11, 3, 1
34, 130, 162, 34, -74, -42, 6, 6, 0
55, 235, 344, 128, -146, -130, 0, 24, 3, -1

Cf. Columns: A000045, A001629, A129707.

Diagonals: A010892, A099254, Antidiagonal sums: A158943.

G.f.: 1/(1-(1+y)*x+(y+1)*(y-1)*x^2).
T(n,0) = A000045(n+1).
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-2,k-2) with T(0,0)= 1 and T(n,k)= 0 if nSum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A090591(n), (-1)^n*A106852(n), A000007(n), A000045(n+1), A000079(n), A057083(n), A190966(n+1) for n = -3, -2, -1, 0, 1, 2, 3 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A010892(n), A000079(n), A030195(n+1), A180222(n+2) for x = 0, 1, 2, 3 respectively.

A202551 Triangle T(n,k), read by rows, given by (1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, -1, 0, -1, 1, -1, 1, 1, -1, -1, 3, -2, -1, 1, 0, 2, -5, 3, 1, -1, 1, -2, -2, 7, -4, -1, 1, 1, -5, 7, 1, -9, 5, 1, -1, 0, -3, 12, -15, 1, 11, -6, -1, 1, -1, 3, 3, -21, 26, -4, -13, 7, 1, -1

Offset: 0

Philippe Deléham, Dec 21 2011

Riordan array (1/(1-x+x^2), x*(x-1)/(1-x+x^2)).

			Triangle begins :
1
1, -1
0, -1, 1
-1, 1, 1, -1
-1, 3, -2, -1, 1
0, 2, -5, 3, 1, -1

Cf. A129267, A199324

T(n,k) = T(n-1,k) + T(n-2,k-1) - T(n-1,k-1) - T(n-2,k).
G.f.: 1/(1+(y-1)*x+(1-y)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A190873(n+1), A190871(n+1), A057086(n), A057085(n+1), A057084(n), A030240(n), A030192(n), A030191(n), A001787(n+1), A057083(n), A099087(n), A010892(n), A000007(n), (-1)^n*A000045(n+1) for x = -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2 respectively.

A227430 Expansion of x^2(1-x)^3/((1-2x)(1-x+x^2)(1-3*x+3x^2)).

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 15, 21, 29, 45, 90, 220, 561, 1365, 3095, 6555, 13110, 25126, 46971, 87381, 164921, 320001, 640002, 1309528, 2707629, 5592405, 11450531, 23166783, 46333566, 91869970, 181348455, 357913941, 708653429, 1410132405, 2820264810, 5662052980

Offset: 0

Paul Curtz, Jul 11 2013

Consider the binomial transform of 0, 0, 0, 0, 0, 1 (period 6) with its differences:
0, 0, 0, 0, 0,  1,  6, 21, 56, 126,...    d(n): after 0, it is A192080.
0, 0, 0, 0, 1,  5, 15, 35, 70, 126,...    e(n)
0, 0, 0, 1, 4, 10, 20, 35, 56,  85,...    f(n)
0, 0, 1, 3, 6, 10, 15, 21, 29,  45,...    a(n)
0, 1, 2, 3, 4,  5,  6,  8, 16,  45,...    b(n)
1, 1, 1, 1, 1,  1,  2,  8, 29,  85,...    c(n)
0, 0, 0, 0, 0,  1,  6, 21, 56, 126,...    d(n).
a(n) + d(n) = A024495(n),
b(n) + e(n) = A131708(n),
c(n) + f(n) = A024493(n).
a(n) - d(n) =  0,  0,  1,  3,  6,  9,   9,   0,...    A057083(n-2)
b(n) - e(n) =  0,  1,  2,  3,  3,  0,  -9, -27,...    A057682(n)
c(n) - f(n) =  1,  1,  1,  0, -3, -9, -18, -27,...    A057681(n)
d(n) - a(n) =  0,  0, -1, -3, -6, -9,  -9,   0,...   -A057083(n-2)
e(n) - b(n) =  0, -1, -2, -3, -3,  0,   9,  27,...   -A057682(n)
f(n) - c(n) = -1, -1, -1,  0,  3,  9,  18,  27,...   -A057681(n).
The first column is A131531(n).
The first two trisections are multiples of 3. Is the third (1, 10, 29,...) mod 9  A029898(n)?

			a(6)=6*10-15*6+20*3-15*1+6*0=15, a(7)=90-150+120-45+6=21.

Seiichi Manyama, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6).

Join[{0},LinearRecurrence[{6,-15,20,-15,6},{0,1,3,6,10},40]] (* Harvey P. Dale, Dec 17 2014 *)

{a(n) = sum(k=0, n\6, binomial(n, 6*k+2))} \\ Seiichi Manyama, Mar 23 2019

a(n) = 6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) for n>5, a(0)=a(1)=0, a(2)=1, a(3)=3, a(4)=6, a(5)=10.
a(n) = A024495(n) - A192080(n-5) for n>4.
G.f.: -(x^5 - 3*x^4 + 3*x^3 - x^2)/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)). - Ralf Stephan, Jul 13 2013
a(n) = Sum_{k=0..floor(n/6)} binomial(n,6*k+2). - Seiichi Manyama, Mar 23 2019

Definition uses the g.f. of Ralf Stephan.

More terms from Harvey P. Dale, Dec 17 2014

A100240 G.f. A(x) satisfies: 4^n/2 = Sum_{k=0..n} [x^k]A(x)^n and also satisfies: ((4+z)^n + z^n)/2 = Sum_{k=0..n} [x^k](A(x)+z*x)^n for all z, where [x^k]A(x)^n denotes the coefficient of x^k in A(x)^n.

Original entry on oeis.org

1, 1, 2, 2, 0, -4, -6, 2, 22, 30, -26, -154, -172, 288, 1190, 990, -3040, -9620, -4970, 31350, 79120, 12580, -318210, -649610, 174150, 3185686, 5233514, -4273078, -31452228, -40495600, 64593386, 305819154, 290278982, -835918098, -2921409370, -1771072346, 9995237616, 27317409988

Offset: 0

Paul D. Hanna, Nov 30 2004

sign

			From the table of powers of A(x), we see that
4^n/2 = Sum of coefficients [x^0] through [x^n] in A(x)^n:
A^1=[1,1],2,2,0,-4,-6,2,22,30,-26,...
A^2=[1,2,5],8,8,0,-16,-24,8,88,120,...
A^3=[1,3,9,19],30,30,2,-54,-84,20,288,...
A^4=[1,4,14,36,73],112,112,16,-176,-288,32,...
A^5=[1,5,20,60,145,281],420,420,90,-570,-988,...
A^6=[1,6,27,92,255,582,1085],1584,1584,440,-1848,...
A^7=[1,7,35,133,413,1071,2331,4201],6006,6006,2002,...
A^8=[1,8,44,184,630,1816,4460,9320,16305],22880,22880,...
the main diagonal of which is:
[x^n]A(x)^(n+1) = (n+1)*A081696(n) for n>=0.

Cf. A100223, A057083, A007440, A081696.

CoefficientList[Series[2*x + Sqrt[1 - 2*x + 5*x^2], {x, 0, 40}], x] (* Vaclav Kotesovec, Feb 07 2021 *)

a(n)=if(n==0,1,(4^n/2-sum(k=0,n,polcoeff(sum(j=0,min(k,n-1),a(j)*x^j)^n+x*O(x^k),k)))/n)

a(n)=polcoeff(2*x+sqrt(1-2*x+5*x^2+x^2*O(x^n)),n)

G.f.: A(x) = 2*x+sqrt(1-2*x+5*x^2).

Recurrence: n*a(n) = (2*n-3)*a(n-1) - 5*(n-3)*a(n-2). - Vaclav Kotesovec, Feb 07 2021

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cp's OEIS Frontend

A115052 Expansion of 1/(3*x^2 - 3*x + 1)^2.

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A134581 a(n) = 4*a(n-1) - 7*a(n-2) + 6*a(n-3) - 3*a(n-4), starting with 0, 1, 2, 3.

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A167925 Triangle, T(n, k) = (sqrt(k+1))^(n-1)*ChebyshevU(n-1, sqrt(k+1)/2), read by rows.

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A194084 Triangle read by rows: a(n)=A135929(n) + A192011(n). Row n gives coefficients of polynomials BC(n,x) in order of decreasing exponents.

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A307395 Expansion of 1/((1 - x) * ((1 - x)^3 + x^3)).

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PARI

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A099447 An Alexander sequence for the knot 6_3.

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A201947 Triangle T(n,k), read by rows, given by (1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,-1,1,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

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A202551 Triangle T(n,k), read by rows, given by (1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A227430 Expansion of x^2*(1-x)^3/((1-2*x)*(1-x+x^2)*(1-3*x+3x^2)).

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A115052 Expansion of 1/(3x^2 - 3x + 1)^2.

A134581 a(n) = 4a(n-1) - 7a(n-2) + 6a(n-3) - 3a(n-4), starting with 0, 1, 2, 3.

A227430 Expansion of x^2(1-x)^3/((1-2x)(1-x+x^2)(1-3*x+3x^2)).