A087455 -id:A087455 - cp's OEIS frontend

A098158 Triangle T(n,k) with diagonals T(n,n-k) = binomial(n, 2*k).

1, 0, 1, 0, 1, 1, 0, 0, 3, 1, 0, 0, 1, 6, 1, 0, 0, 0, 5, 10, 1, 0, 0, 0, 1, 15, 15, 1, 0, 0, 0, 0, 7, 35, 21, 1, 0, 0, 0, 0, 1, 28, 70, 28, 1, 0, 0, 0, 0, 0, 9, 84, 126, 36, 1, 0, 0, 0, 0, 0, 1, 45, 210, 210, 45, 1, 0, 0, 0, 0, 0, 0, 11, 165, 462, 330, 55, 1, 0, 0, 0, 0, 0, 0, 1, 66, 495, 924

Offset: 0

Paul Barry, Aug 29 2004

Row sums are A011782. Inverse is A065547.
Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 1, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jul 29 2006
Sum of entries in column k is A001519(k+1) (the odd-indexed Fibonacci numbers). - Philippe Deléham, Dec 02 2008
Number of permutations of length n avoiding simultaneously the patterns 123 and 132 with k left-to-right minima. A left-to-right minimum in a permutation a(1)a(2)...a(n) is position i such that a(j) > a(i) for all j < i. - Tian Han, Nov 16 2023

			Rows begin
  1;
  0, 1;
  0, 1, 1;
  0, 0, 3, 1;
  0, 0, 1, 6, 1;

G. C. Greubel, Rows n = 0..49 of triangle, flattened
D. Dumont and J. Zeng, Polynomes d'Euler et les fractions continues de Stieltjes-Rogers, Ramanujan J. 2 (1998) 3, 387-410.
Tian Han and Sergey Kitaev, Joint distributions of statistics over permutations avoiding two patterns of length 3, arXiv:2311.02974 [math.CO], 2023.

Cf. A098157, A034839.

Cf. A119900. - Philippe Deléham, Dec 02 2008

Flat(List([0..12], n-> List([0..n], k-> Binomial(n, 2*(n-k)) ))); # G. C. Greubel, Aug 01 2019

[Binomial(n, 2*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 01 2019

Table[Binomial[n, 2*(n-k)], {n,0,12}, {k,0,n}]//Flatten (* Michael De Vlieger, Oct 12 2016 *)

{T(n,k)=polcoeff(polcoeff((1-x*y)/((1-x*y)^2-x^2*y)+x*O(x^n), n, x) + y*O(y^k),k,y)} (Hanna)

T(n,k) = binomial(n, 2*(n-k));
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Aug 01 2019

[[binomial(n, 2*(n-k)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019

T(n,k) = binomial(n,2*(n-k)).
From Tom Copeland, Oct 10 2016: (Start)
E.g.f.: exp(t*x) * cosh(t*sqrt(x)).
O.g.f.: (1/2) * ( 1 / (1 - (1 + sqrt(1/x))*x*t) + 1 / (1 - (1 - sqrt(1/x))*x*t) ).
Row polynomial: x^n * ((1 + sqrt(1/x))^n + (1 - sqrt(1/x))^n) / 2. (End)
Column k is generated by the polynomial Sum_{j=0..floor(k/2)} C(k, 2j) * x^(k-j). - Paul Barry, Jan 22 2005
G.f.: (1-x*y)/((1-x*y)^2 - x^2*y). - Paul D. Hanna, Feb 25 2005
Sum_{k=0..n} x^k*T(n,k)= A009116(n), A000007(n), A011782(n), A006012(n), A083881(n), A081335(n), A090139(n), A145301(n), A145302(n), A145303(n), A143079(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Dec 04 2006, Oct 15 2008, Oct 19 2008
T(n,k) = T(n-1,k-1) + Sum_{i=0..k-1} T(n-2-i,k-1-i); T(0,0)=1; T(n,k)=0 if n < 0 or k < 0 or n < k. E.g.: T(8,5) = T(7,4) + T(6,4) + T(5,3) + T(4,2) + T(3,1) + T(2,0) = 7+15+5+1+0+0 = 28. - Philippe Deléham, Dec 04 2006
Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A011782(n), A001333(n), A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 respectively. - Philippe Deléham, Dec 24 2007
Sum_{k=0..n} T(n,k)*(-x)^(n-k) = A000012(n), A146559(n), A087455(n), A138230(n), A006495(n), A138229(n) for x = 0,1,2,3,4,5 respectively. - Philippe Deléham, Nov 14 2008
T(n,k) = A085478(k,n-k). - Philippe Deléham, Dec 02 2008
T(n,k) = 2*T(n-1,k-1) + T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(1,1) = 1, T(1,0) = 0 and T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Mar 15 2012

A028297 Coefficients of Chebyshev polynomials of the first kind: triangle of coefficients in expansion of cos(n*x) in descending powers of cos(x).

Original entry on oeis.org

1, 1, 2, -1, 4, -3, 8, -8, 1, 16, -20, 5, 32, -48, 18, -1, 64, -112, 56, -7, 128, -256, 160, -32, 1, 256, -576, 432, -120, 9, 512, -1280, 1120, -400, 50, -1, 1024, -2816, 2816, -1232, 220, -11, 2048, -6144, 6912, -3584, 840, -72, 1, 4096, -13312, 16640, -9984

Offset: 0

N. J. A. Sloane

Rows are of lengths 1, 1, 2, 2, 3, 3, ... (A008619).
This triangle is generated from A118800 by shifting down columns to allow for (1, 1, 2, 2, 3, 3, ...) terms in each row. - Gary W. Adamson, Dec 16 2007
Unsigned triangle = A034839 * A007318. - Gary W. Adamson, Nov 28 2008
Triangle, with zeros omitted, given by (1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, -1, 1, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 16 2011
From Wolfdieter Lang, Aug 02 2014: (Start)
This irregular triangle is the row reversed version of the Chebyshev T-triangle A053120 given by A039991 with vanishing odd-indexed columns removed.
If zeros are appended in each row n >= 1, in order to obtain a regular triangle (see the Philippe Deléham comment, g.f. and example) this becomes the Riordan triangle (1-x)/(1-2*x), -x^2/(1-2*x). See also the unsigned version A201701 of this regular triangle.
(End)
Apparently, unsigned diagonals of this array are rows of A200139. - Tom Copeland, Oct 11 2014
It appears that the coefficients are generated by the following: Let SM_k = Sum( d_(t_1, t_2)* eM_1^t_1 * eM_2^t_2) summed over all length 2 integer partitions of k, i.e., 1*t_1 + 2*t_2 = k, where SM_k are the averaged k-th power sum symmetric polynomials in 2 data (i.e., SM_k = S_k/2 where S_k are the k-th power sum symmetric polynomials, and where eM_k are the averaged k-th elementary symmetric polynomials, eM_k = e_k/binomial(2,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2) form an irregular triangle, with one row for each k value starting with k=1. Thus this procedure and associated OEIS sequences A287768, A288199, A288207, A288211, A288245, A288188 are generalizations of Chebyshev polynomials of the first kind. - Gregory Gerard Wojnar, Jul 01 2017

			Letting c = cos x, we have: cos 0x = 1, cos 1x = 1c; cos 2x = 2c^2-1; cos 3x = 4c^3-3c, cos 4x = 8c^4-8c^2+1, etc.
T4 = 8x^4 - 8x^2 + 1 = 8, -8, +1 = 2^(3) - (4)(2) + [2^(-1)](4)/2.
From _Wolfdieter Lang_, Aug 02 2014: (Start)
The irregular triangle T(n,k) begins:
n\k     1      2     3      4     5     6   7   8 ....
0:      1
1:      1
2:      2     -1
3:      4     -3
4:      8     -8     1
5:     16    -20     5
6:     32    -48    18     -1
7:     64   -112    56     -7
8:    128   -256   160    -32     1
9:    256   -576   432   -120     9
10:   512  -1280  1120   -400    50    -1
11:  1024  -2816  2816  -1232   220   -11
12:  2048  -6144  6912  -3584   840   -72   1
13:  4096 -13312 16640  -9984  2912  -364  13
14:  8192 -28672 39424 -26880  9408 -1568  98  -1
15: 16384 -61440 92160 -70400 28800 -6048 560 -15
...
T(4,x) = 8*x^4 -8*x^2 + 1*x^0, T(5,x) = 16*x^5 - 20*x^3 + 5*x^1, with Chebyshev's T-polynomials (A053120). (End)
From _Philippe Deléham_, Dec 16 2011: (Start)
The triangle (1,1,0,0,0,0,...) DELTA (0,-1,1,0,0,0,0,...) includes zeros and begins:
   1;
   1,   0;
   2,  -1,  0;
   4,  -3,  0,  0;
   8,  -8,  1,  0, 0;
  16, -20,  5,  0, 0, 0;
  32, -48, 18, -1, 0, 0, 0; (End)

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 5th ed., Section 1.335, p. 35.
S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 106. [From Rick L. Shepherd, Jul 06 2010]

Alois P. Heinz, Rows n = 0..200, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy] p. 795.
Pantelis A. Damianou, A Beautiful Sine Formula, Amer. Math. Monthly 121 (2014), no. 2, 120--135. MR3149030.
Daniel J. Greenhoe, Frames and Bases: Structure and Design, Version 0.20, Signal Processing ABCs series (2019) Vol. 4, see page 172.
Daniel J. Greenhoe, A Book Concerning Transforms, Version 0.10, Signal Processing ABCs series (2019) Vol. 5, see page 94.
Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, and Minghao Chen, Recursive sequences and Girard-Waring identities with applications in sequence transformation, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062.
C. Lanczos, Applied Analysis (Annotated scans of selected pages)
G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, Table GW.n=2, p. 22, arXiv:1706.08381 [math.GM], 2017.

Cf. A028298.
Reflection of A008310, the main entry. With zeros: A039991.
Cf. A053120 (row reversed table including zeros).
Cf. A118800, A034839, A081277, A124182.
Cf. A001333 (row sums 1), A001333 (alternating row sums). - Wolfdieter Lang, Aug 02 2014

b:= proc(n) b(n):= `if`(n<2, 1, expand(2*b(n-1)-x*b(n-2))) end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
seq(T(n), n=0..15);  # Alois P. Heinz, Sep 04 2019

t[n_] := (Cos[n x] // TrigExpand) /. Sin[x]^m_ /; EvenQ[m] -> (1 - Cos[x]^2)^(m/2) // Expand; Flatten[Table[ r = Reverse @ CoefficientList[t[n], Cos[x]]; If[OddQ[Length[r]], AppendTo[r,0]]; Partition[r,2][[All, 1]],{n, 0, 13}] ][[1 ;; 53]] (* Jean-François Alcover, May 06 2011 *)
Tpoly[n_] := HypergeometricPFQ[{(1 - n)/2, -n/2}, {1/2}, 1 - x];
Table[CoefficientList[Tpoly[n], x], {n, 0, 12}] // Flatten (* Peter Luschny, Feb 03 2021 *)

cos(n*x) = 2 * cos((n-1)*x) * cos(x) - cos((n-2)*x) (from CRC's Multiple-angle relations). - Rick L. Shepherd, Jul 06 2010
G.f.: (1-x) / (1-2x+y*x^2). - Philippe Deléham, Dec 16 2011
Sum_{k=0..n} T(n,k)*x^k = A011782(n), A000012(n), A146559(n), A087455(n), A138230(n), A006495(n), A138229(n) for x = 0, 1, 2, 3, 4, 5, 6, respectively. - Philippe Deléham, Dec 16 2011
T(n,k) = [x^k] hypergeom([1/2 - n/2, -n/2], [1/2], 1 - x). - Peter Luschny, Feb 03 2021
T(n,k) = (-1)^k * 2^(n-1-2*k) * A034807(n,k). - Hoang Xuan Thanh, Jun 21 2025

More terms from David W. Wilson

Row length sequence and link to Abramowitz-Stegun added by Wolfdieter Lang, Aug 02 2014

A088137 Generalized Gaussian Fibonacci integers.

Original entry on oeis.org

0, 1, 2, 1, -4, -11, -10, 13, 56, 73, -22, -263, -460, -131, 1118, 2629, 1904, -4079, -13870, -15503, 10604, 67717, 103622, 4093, -302680, -617639, -327238, 1198441, 3378596, 3161869, -3812050, -17109707, -22783264, 5762593, 79874978, 142462177, 45299420, -336787691

Offset: 0

Paul Barry, Sep 20 2003

The Lucas U(P=2, Q=3) sequence. - R. J. Mathar, Oct 24 2012
Hence for n >= 0, a(n+2)/a(n+1) equals the continued fraction 2 - 3/(2 - 3/(2 - 3/(2 - ... - 3/2))) with n 3's. - Greg Dresden, Oct 06 2019
With different signs, 0, 1, -2, 1, 4, -11, 10, 13, -56, 73, 22, -263, 460, ... also the Lucas U(-2,3) sequence. - R. J. Mathar, Jan 08 2013
From Peter Bala, Apr 01 2018: (Start)
The companion Lucas sequence V(n,2,3) is A087455.
Define a binary operation o on rational numbers by x o y = (x + y)/(1 - 2*x*y). This is a commutative and associative operation with identity 0. Then 1 o 1 o ... o 1 (n terms) = a(n)/A087455(n). Cf. A025172 and A127357. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
Ronald Orozco López, Deformed Differential Calculus on Generalized Fibonacci Polynomials, arXiv:2211.04450 [math.CO], 2022.
Mihai Prunescu, On other two representations of the C-recursive integer sequences by terms in modular arithmetic, arXiv:2406.06436 [math.NT], 2024. See p. 18.
Mihai Prunescu and Lorenzo Sauras-Altuzarra, On the representation of C-recursive integer sequences by arithmetic terms, arXiv:2405.04083 [math.LO], 2024. See p. 16.
Mihai Prunescu and Joseph M. Shunia, On modular representations of C-recursive integer sequences, arXiv:2502.16928 [math.NT], 2025. See p. 6.
Wikipedia, Lucas sequence
Index entries for linear recurrences with constant coefficients, signature (2,-3).
Index entries for Lucas sequences

Cf. A084102, A088138, A045873, A088139.

Cf. A087455, A025172, A123562, A127357.

[n le 2 select n-1 else 2*Self(n-1)-3*Self(n-2): n in [1..50]]; // G. C. Greubel, Oct 22 2018

A[0]:= 0: A[1]:= 1:
for n from 2 to 100 do A[n]:= 2*A[n-1] - 3*A[n-2] od:
seq(A[n],n=0..100); # Robert Israel, Aug 05 2014

LinearRecurrence[{2,-3},{0,1},40] (* Harvey P. Dale, Nov 03 2014 *)

x='x+O('x^50); concat([0], Vec(x/(1-2*x+3*x^2))) \\ G. C. Greubel, Oct 22 2018

[lucas_number1(n,2,3) for n in range(0, 38)] # Zerinvary Lajos, Apr 23 2009

a(n) = 3^(n/2)*sin(n*atan(sqrt(2)))/sqrt(2).
|3*A087455(n) - A087455(n+1)| = 2*a(n+1) or 3*A087455(n) + A087455(n+1) = 2*a(n+1). - Creighton Dement, Aug 02 2004
G.f.: x/(1 - 2*x + 3*x^2).
E.g.f.: exp(x)*sin(sqrt(2)*x)/sqrt(2).
a(n) = 2*a(n-1) - 3*a(n-2) for n > 1, a(0)=0, a(1)=1.
a(n) = ((1 + i*sqrt(2))^n - (1 - i*sqrt(2))^n)/(2*i*sqrt(2)), where i=sqrt(-1).
a(n) = Im((1 + i*sqrt(2))^n/sqrt(2)).
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k+1)(-2)^k.
3^(n+1) = 9*(A087455(n))^2 + 2*(A087455(n+1))^2 - 2*(a(n+2))^2; 3^n = a(n+1)^2 + 3*a(n)^2 - 2*a(n+1)*a(n) for n > 0 - Creighton Dement, Jan 20 2005
G.f.: G(0)*x/(2*(1-x)), where G(k) = 1 + 1/(1 - x*(2*k+1)/(x*(2*k+3) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 25 2013
G.f.: Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(4*k+2 - 3*x)/( x*(4*k+4 - 3*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 06 2013
a(n+1) = Sum_{k=0..n} A123562(n,k). - Philippe Deléham, Nov 23 2013
a(n) = n*hypergeom([(1-n)/2,(2-n)/2],[3/2],-2). - Gerry Martens, Sep 03 2023

A124182 A skewed version of triangular array A081277.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 0, 3, 4, 0, 0, 1, 8, 8, 0, 0, 0, 5, 20, 16, 0, 0, 0, 1, 18, 48, 32, 0, 0, 0, 0, 7, 56, 112, 64, 0, 0, 0, 0, 1, 32, 160, 256, 128, 0, 0, 0, 0, 0, 9, 120, 432, 576, 256, 0, 0, 0, 0, 0, 1, 50, 400, 1120, 1280, 512

Offset: 0

Philippe Deléham, Dec 05 2006

Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, -1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 0, 0, 0, 0, 0, 0,...] where DELTA is the operator defined in A084938. Falling diagonal sums in A052980.

			Triangle begins:
  1;
  0, 1;
  0, 1, 2;
  0, 0, 3, 4;
  0, 0, 1, 8,  8;
  0, 0, 0, 5, 20, 16;
  0, 0, 0, 1, 18, 48,  32;
  0, 0, 0, 0,  7, 56, 112,  64;
  0, 0, 0, 0,  1, 32, 160, 256,  128;
  0, 0, 0, 0,  0,  9, 120, 432,  576,  256;
  0, 0, 0, 0,  0,  1,  50, 400, 1120, 1280, 512;

Cf. A025192 (column sums). Diagonals include A011782, A001792, A001793, A001794, A006974, A006975, A006976.

T(0,0)=T(1,1)=1, T(n,k)=0 if n < k or if k < 0, T(n,k) = T(n-2,k-1) + 2*T(n-1,k-1).
Sum_{k=0..n} x^k*T(n,k) = (-1)^n*A090965(n), (-1)^n*A084120(n), (-1)^n*A006012(n), A033999(n), A000007(n), A001333(n), A084059(n) for x = -4, -3, -2, -1, 0, 1, 2 respectively.
Sum_{k=0..floor(n/2)} T(n-k,k) = Fibonacci(n-1) = A000045(n-1).
Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A011782(n), A001333(n), A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 respectively. - Philippe Deléham, Dec 26 2007
Sum_{k=0..n} T(n,k)*(-x)^(n-k) = A011782(n), A000012(n), A146559(n), A087455(n), A138230(n), A006495(n), A138229(n) for x= 0,1,2,3,4,5,6 respectively. - Philippe Deléham, Nov 14 2008
G.f.: (1-y*x)/(1-2y*x-y*x^2). - Philippe Deléham, Dec 04 2011
Sum_{k=0..n} T(n,k)^2 = A002002(n) for n > 0. - Philippe Deléham, Dec 04 2011

A077966 Expansion of 1/(1+2*x^2).

Original entry on oeis.org

1, 0, -2, 0, 4, 0, -8, 0, 16, 0, -32, 0, 64, 0, -128, 0, 256, 0, -512, 0, 1024, 0, -2048, 0, 4096, 0, -8192, 0, 16384, 0, -32768, 0, 65536, 0, -131072, 0, 262144, 0, -524288, 0, 1048576, 0, -2097152, 0, 4194304, 0, -8388608, 0, 16777216, 0, -33554432, 0, 67108864, 0, -134217728, 0, 268435456

Offset: 0

N. J. A. Sloane, Nov 17 2002

Normally sequences like this are not included, since with the alternating 0's deleted it is already in the database.
Inverse binomial transform of A087455. - Philippe Deléham, Dec 02 2008
Pisano period lengths: 1, 1, 2, 1, 8, 2, 12, 1, 6, 8, 10, 2, 24, 12, 8, 1, 16, 6, 18, 8,... - R. J. Mathar, Aug 10 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,-2).

Cf. A000079, A077957.

a:=[1,0];; for n in [3..60] do a[n]:=-2*a[n-2]; od; a; # G. C. Greubel, Jun 24 2019

I:=[1,0]; [n le 2 select I[n] else -2*Self(n-2): n in [1..60]]; // G. C. Greubel, Jun 24 2019

A077966:=n->(1+(-1)^n)*(-2)^(n/2)/2; seq(A077966(n), n=0..50); # Wesley Ivan Hurt, Apr 02 2014

CoefficientList[Series[1/(1 + 2*x^2), {x,0,60}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
LinearRecurrence[{0,-2}, {1,0}, 60] (* G. C. Greubel, Jun 24 2019 *)

Vec(1/(1+2*x^2)+O(x^60)) \\ Charles R Greathouse IV, Sep 24 2012

for(n=0, 51, print1(imag(quadgen(-8)^(n+1)), ", ")) \\ Arkadiusz Wesolowski, Dec 26 2012

[lucas_number1(n,0,2) for n in range(1,60)] # Zerinvary Lajos, Jul 16 2008

a(n) = (1+(-1)^n)*(-2)^(n/2)/2. - R. J. Mathar, Apr 23 2009
a(n) = ((n+1) mod 2 )*(-2)^floor((n+1)/2). - Wesley Ivan Hurt, Apr 06 2014
E.g.f.: cos(sqrt(2)*x). - G. C. Greubel, Jun 24 2019

A127357 Expansion of 1/(1 - 2x + 9x^2).

Original entry on oeis.org

1, 2, -5, -28, -11, 230, 559, -952, -6935, -5302, 51811, 151340, -163619, -1689298, -1906025, 11391632, 39937489, -22649710, -404736821, -605626252, 2431378885, 10313394038, -1255621889

Offset: 0

Paul Barry, Jan 11 2007

Hankel transform of A100193. A member of the family of sequences with g.f. 1/(1-2*x+r^2*x^2) which are the Hankel transforms of the sequences given by Sum_{k=0..n} binomial(2*n,k)*r^(n-k).
From Peter Bala, Apr 01 2018: (Start)
With offset 1, this is the Lucas sequence U(n,2,9). The companion Lucas sequence V(n,2,9) is 2*A025172(n).
Define a binary operation o on rational numbers by x o y = (x + y)/(1 - 2*x*y). This is a commutative and associative operation with identity 0. Then 2 o 2 o ... o 2 (n terms) = 2*A127357(n-1)/A025172(n). Cf. A088137 and A087455. (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Wikipedia, Lucas sequence
Index entries for linear recurrences with constant coefficients, signature (2,-9).

Variant is A025170.

Cf. A025172, A087455, A088137, A100193, A127357.

a:=[1,2];; for n in [3..25] do a[n]:=2*a[n-1]-9*a[n-2]; od; a; # Muniru A Asiru, Oct 23 2018

m:=23; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-2*x+9*x^2))); // Bruno Berselli, Jul 01 2011

[3^n*Evaluate(ChebyshevU(n+1),1/3): n in [0..50]]; // G. C. Greubel, Jan 02 2024

c := 2*sqrt(2): g := exp(x)*(sin(c*x)+c*cos(c*x))/c: ser := series(g,x,32):
seq(n!*coeff(ser,x,n), n=0..22); # Peter Luschny, Oct 19 2016

RootReduce@Table[3^n (Cos[n ArcTan[2 Sqrt[2]]] + Sin[n ArcTan[2 Sqrt[2]]] Sqrt[2]/4), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 15 2016 *)
CoefficientList[Series[1/(1-2x+9x^2),{x,0,40}],x] (* or *)
LinearRecurrence[ {2,-9},{1,2},40] (* Harvey P. Dale, Mar 15 2022 *)
Table[3^n*ChebyshevU[n, 1/3], {n,0,40}] (* G. C. Greubel, Jan 02 2024 *)

makelist(coeff(taylor(1/(1-2*x+9*x^2), x, 0, n), x, n), n, 0, 22); /* Bruno Berselli, Jul 01 2011 */

Vec(1/(1-2*x+9*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

[lucas_number1(n,2,9) for n in range(1, 24)] # Zerinvary Lajos, Apr 23 2009

[3^n*chebyshev_U(n,1/3) for n in range(41)] # G. C. Greubel, Jan 02 2024

a(n) = Sum_{k=0..n} binomial(n-k,k)*2^(n-2*k)*(-9)^k.
a(n) = 2*a(n-1) - 9*a(n-2) for n >= 2. - Vincenzo Librandi, Mar 22 2011
a(n) = ((1-2*sqrt(2)*i)^n-(1+2*sqrt(2)*i)^n)*i/(4*sqrt(2)), where i=sqrt(-1). - Bruno Berselli, Jul 01 2011
From Vladimir Reshetnikov, Oct 15 2016: (Start)
a(n) = 3^n*(cos(n*theta) + sin(n*theta)*sqrt(2)/4), theta = arctan(2*sqrt(2)).
E.g.f.: exp(x)*(cos(2*sqrt(2)*x) + sin(2*sqrt(2)*x)*sqrt(2)/4). (End)
a(n) = 2^n*Product_{k=1..n}(1 + 3*cos(k*Pi/(n+1))). - Peter Luschny, Nov 28 2019
From G. C. Greubel, Jan 02 2024: (Start)
a(n) = (-1)^n * A025170(n).
a(n) = 3^n * ChebyshevU(n, 1/3). (End)

A168175 Expansion of 1/(1 - 4x + 7x^2).

Original entry on oeis.org

1, 4, 9, 8, -31, -180, -503, -752, 513, 7316, 25673, 51480, 26209, -255524, -1205559, -3033568, -3695359, 6453540, 51681673, 161551912, 284435937, 6880364, -1963530103, -7902282960, -17864421119, -16141703756, 60484132809

Offset: 0

Roger L. Bagula, Nov 19 2009

Also the coefficient of i of Q^(n+1), Q being the quaternion 2+i+j+k. The real part of the quaternion power is A213421, see also A087455, A088138, A128018. - Stanislav Sykora, Jun 11 2012

a(n)*(-1)^n gives the coefficient c(7^n) of (eta(z^6))^4, a modular cusp form of weight 2, when expanded in powers of q = exp(2*Pi*i*z), Im(z) > 0, assuming alpha-multiplicativity (but not for primes 2 and 3) with alpha(x) = x (weight 2) and input c(7) = -4. Eta is the Dedekind function. See the Apostol reference, p. 138, eq. (54) for alpha-multiplicativity and p. 130, eq. (39) with k=2. See also A000727(n) = b(n) where c(7^n) = b((7^n-1)/6) = b(A023000(n)), n >= 0. Proof: The alpha-multiplicity with alpha(1) = 1 and c(1) = 1 leads from p^n = p^(n-1)*p to the recurrence c_n = c*c_(n-1) - a*c(n-2), with c_n = c(p^n), c = c(p) and a = alpha(p). Inputs are c_{-1} = 0 and c_0 = c(1) = 1. This gives the polynomial c_n = sqrt(a)^n * S(n,c/sqrt(a)), with Chebyshev's S-polynomials (A049310). See the Apostol reference, Exercise 6., p. 139. Here p = 7, c = -4. - Wolfdieter Lang, Apr 27 2016

			G.f. = 1 + 4*x + 9*x^2 + 8*x^3 - 31*x^4 - 180*x^5 - 503*x^6 - 752*x^7 + ... - _Michael Somos_, Feb 23 2020

Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990, pp. 130, 138 - 139.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-7).
Index entries for sequences related to Chebyshev polynomials.

Cf. A023000, A049310.

I:=[1,4]; [n le 2 select I[n] else 4*Self(n-1)-7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 25 2012

CoefficientList[Series[1/(1-4x+7x^2),{x,0,30}],x] (* or *) LinearRecurrence[ {4,-7},{1,4},30] (* Harvey P. Dale, Nov 28 2014 *)

{a(n) = my(s=1, t=1); if( n<0, n=-2-n; s=-1; t=1/7); s * t^(n+1) * polcoeff(1 / (1 - 4*x + 7*x^2) + x * O(x^n), n)}; /* Michael Somos, Feb 23 2020 */

a(n) = (1/2 - i/sqrt(3))*(2 + i*sqrt(3))^n + (1/2 + i/sqrt(3))*(2 - i*sqrt(3))^n (Binet formula), where i is the imaginary unit.
a(n) = 4*a(n-1) - 7*a(n-2).
a(n) = sqrt(7)^n * S(n, 4/sqrt(7)), n >= 0, with Chebyshev's S polynomials (A049310). - Wolfdieter Lang, Apr 27 2016
E.g.f.: (2*sqrt(3)*sin(sqrt(3)*x) + 3*cos(sqrt(3)*x))*exp(2*x)/3. - Ilya Gutkovskiy, Apr 27 2016
a(n) = (-1) * 7^(n+1) * a(-2-n) for all n in Z. - Michael Somos, Feb 23 2020

A201701 Riordan triangle ((1-x)/(1-2x), x^2/(1-2x)).

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 4, 3, 0, 0, 8, 8, 1, 0, 0, 16, 20, 5, 0, 0, 0, 32, 48, 18, 1, 0, 0, 0, 64, 112, 56, 7, 0, 0, 0, 0, 128, 256, 160, 32, 1, 0, 0, 0, 0, 256, 576, 432, 120, 9, 0, 0, 0, 0, 0, 512, 1280, 1120, 400, 50, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 03 2011

Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (0,1,-1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.
Skewed version of triangle in A200139.
Triangle without zeros: A207537.
For the version with negative odd numbered columns, which is Riordan ((1-x)/(1-2*x), -x^2/(1-2*x)) see comments on A028297 and A039991. - Wolfdieter Lang, Aug 06 2014
This is an example of a stretched Riordan array in the terminology of Section 2 of Corsani et al. - Peter Bala, Jul 14 2015

			The triangle T(n,k) begins:
  n\k      0     1     2     3     4    5   6  7 8 9 10 11 ...
  0:       1
  1:       1     0
  2:       2     1     0
  3:       4     3     0     0
  4:       8     8     1     0     0
  5:      16    20     5     0     0    0
  6:      32    48    18     1     0    0   0
  7:      64   112    56     7     0    0   0  0
  8:     128   256   160    32     1    0   0  0 0
  9:     256   576   432   120     9    0   0  0 0 0
  10:    512  1280  1120   400    50    1   0  0 0 0  0
  11:   1024  2816  2816  1232   220   11   0  0 0 0  0  0
  ...  reformatted and extended. - _Wolfdieter Lang_, Aug 06 2014

C. Corsani, D. Merlini, and R. Sprugnoli, Left-inversion of combinatorial sums, Discrete Mathematics, 180 (1998) 107-122.

Columns include A011782, A001792, A001793, A001794, A006974, A006975, A006976.
Diagonals sums are in A052980.
Cf. A028297, A081265, A124182, A131577, A039991 (zero-columns deleted, unsigned and zeros appended).
Cf. A098158, A200139, A207537.
Cf. A028297 (signed version, zeros deleted). Cf. A034839.

(* The function RiordanArray is defined in A256893. *)
RiordanArray[(1 - #)/(1 - 2 #)&, #^2/(1 - 2 #)&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

T(n,k) = 2*T(n-1,k) + T(n-2,k-1) with T(0,0) = T(1,0) = 1, T(1,1) = 0 and T(n,k) = 0 for k<0 or for n

Sum_{k=0..n} T(n,k)^2 = A002002(n) for n>0.

Sum_{k=0..n} T(n,k)*x^k = A138229(n), A006495(n), A138230(n), A087455(n), A146559(n), A000012(n), A011782(n), A001333(n), A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17 respectively.

G.f.: (1-x)/(1-2*x-y*x^2). - Philippe Deléham, Mar 03 2012

From Peter Bala, Jul 14 2015: (Start)

Factorizes as A034839 * A007318 = (1/(1 - x), x^2/(1 - x)^2) * (1/(1 - x), x/(1 - x)) as a product of Riordan arrays.

T(n,k) = Sum_{i = k..floor(n/2)} binomial(n,2*i) *binomial(i,k). (End)

Name changed, keyword:easy added, crossrefs A028297 and A039991 added, and g.f. corrected by Wolfdieter Lang, Aug 06 2014

A025172 Let phi = arccos(1/3), the dihedral angle of the regular tetrahedron. Then cos(n*phi) = a(n)/3^n.

Original entry on oeis.org

1, 1, -7, -23, 17, 241, 329, -1511, -5983, 1633, 57113, 99529, -314959, -1525679, -216727, 13297657, 28545857, -62587199, -382087111, -200889431, 3037005137, 7882015153, -11569015927, -94076168231, -84031193119, 678623127841, 2113526993753

Offset: 0

Wouter Meeussen

Used when showing that the regular simplex is not "scisssors-dissectible" to a cube, thus answering Hilbert's third problem.
From Peter Bala, Apr 01 2018: (Start)
This sequence is (1/2) * the Lucas sequence V(n,2,9). The companion Lucas sequence U(n,2,9) is A127357.
Define a binary operation o on rational numbers by x o y = (x + y)/(1 - 2*x*y). This is a commutative and associative operation with identity 0. Then 2 o 2 o ... o 2 (n terms) = 2*A127357(n-1)/A025172(n). Cf. A088137 and A087455. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J. L. Dupont, Scissors Congruences, Group Homology and Characteristic Classes, World Scientific, 2001. See p. 4.
Wikipedia, Lucas sequence
Index entries for linear recurrences with constant coefficients, signature (2,-9).

Cf. A088137, A087455, A127357.

f:=proc(n) option remember; if n <= 1 then RETURN(1); fi; 2*f(n-1)-9*f(n-2); end;

Table[ n/2 3^n GegenbauerC[ n, 1/3 ], {n, 24} ]
CoefficientList[Series[(1 - x)/(1 - 2 x + 9 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 17 2013 *)
LinearRecurrence[{2,-9},{1,1},30] (* Harvey P. Dale, Jan 30 2016 *)

{a(n)= if(n<0, 0, 3^(n-1)* subst(3* poltchebi(abs(n)), x, 1/3))} /* Michael Somos, Mar 14 2007 */

a(0) = 1, a(1) = 1; for n >= 2, a(n) = 2*a(n-1) - 9*a(n-2). - Warut Roonguthai, Oct 11 2005
a(n) = (1/2)*(1-2*i*2^(1/2))^n+(1/2)*(1+2*i*2^(1/2))^n, where i=sqrt(-1). - Vladeta Jovovic, Apr 19 2003
a(n) is the permanent of the matrix M^n, where M = [i, 2; 1, i]. - Simone Severini, Apr 27 2007
a(n) = Product_{i=1..n} (2 - tan((i-1/2)*Pi/(2*n))^2). - Gerry Martens, May 26 2011
G.f.: (1-x)/(1-2*x+9*x^2). - Colin Barker, Jun 21 2012
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(8*k+1)/(x*(8*k+9) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 28 2013
E.g.f.: exp(x)*cos(2*sqrt(2)*x). - Vladimir Reshetnikov, Oct 15 2016
a(n) = A127357(n)-A127357(n-1). - R. J. Mathar, Apr 07 2022

Better description from Vladeta Jovovic, Apr 19 2003

Edited by N. J. A. Sloane, Feb 22 2007. Among other things, I changed the offset and the beginning of the sequence, so some of the formulas may need to be adjusted slightly.

A213421 Real part of Q^n, Q being the quaternion 2+i+j+k.

Original entry on oeis.org

1, 2, 1, -10, -47, -118, -143, 254, 2017, 6290, 11041, 134, -76751, -307942, -694511, -622450, 2371777, 13844258, 38774593, 58188566, -38667887, -561991510, -1977290831, -3975222754, -2059855199, 19587138482

Offset: 0

Stanislav Sykora, Jun 11 2012

Stanislav Sykora, Table of n, a(n) for n = 0..1000
Beata Bajorska-Harapińska, Barbara Smoleń, Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras Vol. 29, No. 3 (2019), Article 54.
Wikipedia, Lucas sequence

Cf. A168175, A087455, A088138, A128018, A146559.

#A213421
seq(simplify(1/2*((2+I*sqrt(3))^n+(2-I*sqrt(3))^n)), n = 0 .. 25); # Peter Bala, Mar 29 2015

QuaternionToN(a,b,c,d,nmax) = {local (C);C = matrix(nmax+1,4);C[1,1]=1;for(n=2,nmax+1,C[n,1]=a*C[n-1,1]-b*C[n-1,2]-c*C[n-1,3]-d*C[n-1,4];C[n,2]=b*C[n-1,1]+a*C[n-1,2]+d*C[n-1,3]-c*C[n-1,4];C[n,3]=c*C[n-1,1]-d*C[n-1,2]+a*C[n-1,3]+b*C[n-1,4];C[n,4]=d*C[n-1,1]+c*C[n-1,2]-b*C[n-1,3]+a*C[n-1,4];);return (C);}
Q=QuaternionToN(2,1,1,1,1000);
for(n=1,#Q[,1],write("A213421.txt",n-1," ",Q[n,1]));

Conjecture: G.f. (1-2x)/(1-4x+7x^2). a(n) = A168175(n)-2*A168175(n-1). - R. J. Mathar, Jun 25 2012
From Peter Bala, Mar 29 2015: (Start)
The above o.g.f. is correct; this is the Lucas sequence V_n(4,7).
a(n) = Re( (2 + sqrt(3)*i)^n )= 1/2*( (2 + sqrt(3)*i)^n + (2 - sqrt(3)*i)^n ).
a(n) = 1/2 * trace( [ 2 + i, 1 + i; -1 + i, 2 - i ]^n ) = 1/2 * trace( [ 2 , sqrt(3)*i ; sqrt(3)*i, 2 ]^n ).
a(n) = 4*a(n-1) - 7*a(n-2) with a(0) = 1, a(1) = 2. (End)

Showing 1-10 of 15 results. Next

cp's OEIS Frontend

A098158 Triangle T(n,k) with diagonals T(n,n-k) = binomial(n, 2*k).

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A028297 Coefficients of Chebyshev polynomials of the first kind: triangle of coefficients in expansion of cos(n*x) in descending powers of cos(x).

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A088137 Generalized Gaussian Fibonacci integers.

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A124182 A skewed version of triangular array A081277.

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A077966 Expansion of 1/(1+2*x^2).

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A127357 Expansion of 1/(1 - 2*x + 9*x^2).

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A127357 Expansion of 1/(1 - 2x + 9x^2).

A168175 Expansion of 1/(1 - 4x + 7x^2).

A201701 Riordan triangle ((1-x)/(1-2x), x^2/(1-2x)).