A138230 -id:A138230 - 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

A088138 Generalized Gaussian Fibonacci integers.

Original entry on oeis.org

0, 1, 2, 0, -8, -16, 0, 64, 128, 0, -512, -1024, 0, 4096, 8192, 0, -32768, -65536, 0, 262144, 524288, 0, -2097152, -4194304, 0, 16777216, 33554432, 0, -134217728, -268435456, 0, 1073741824, 2147483648, 0, -8589934592, -17179869184, 0, 68719476736, 137438953472

Offset: 0

Paul Barry, Sep 20 2003

The sequence 0,1,-2,0,8,-16,... has g.f. x/(1+2*x-4*x^2), a(n) = 2^n*sin(2n*Pi/3)/sqrt(3) and is the inverse binomial transform of sin(sqrt(3)*x)/sqrt(3): 0,1,-3,0,9,...
a(n+1) is the Hankel transform of A100192. - Paul Barry, Jan 11 2007
a(n+1) is the trinomial transform of A010892: a(n+1) = Sum_{k=0..2n} trinomial(n,k)*A010892(k+1) where trinomial(n, k) = trinomial coefficients (A027907). - Paul Barry, Sep 10 2007
a(n+1) is the Hankel transform of A100067. - Paul Barry, Jun 16 2009
From Paul Curtz, Oct 04 2009: (Start)
1) a(n) = A131577(n)*A128834(n).
2) Binomial transform of 0,1,0,-3,0,9,0,-27, see A000244.
3) Sequence is identical to every 2n-th difference divided by (-3)^n.
4) a(3n) + a(3n+1) + a(3n+2) = (-1)^n*3*A001018(n) for n >= 1.
5) For missing terms in a(n) see A013731 = 4*A001018. (End)
The coefficient of i of Q^n, where Q is the quaternion 1+i+j+k. Due to symmetry, also the coefficients of j and of k. - Stanislav Sykora, Jun 11 2012 [The coefficients of 1 are in A138230. - Wolfdieter Lang, Jan 28 2016]
With different signs, 0, 1, -2, 0, 8, -16, 0, 64, -128, 0, 512, -1024, ... is the Lucas U(-2,4) sequence. - R. J. Mathar, Jan 08 2013

Robert Israel, Table of n, a(n) for n = 0..3300
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.
Wikipedia, Lucas sequence
Index entries for linear recurrences with constant coefficients, signature (2,-4).
Index entries for Lucas sequences

Cf. A084102, A088137, A045873, A088139, A138230.

a:=[0,1];; for n in [3..40] do a[n]:=2*a[n-1]-4*a[n-2]; od; a; # Muniru A Asiru, Oct 23 2018

I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1) - 4*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 15 2018

M:= <<1+I,1+I>|>:
T:= <<-I/2,0>|<0,I/2>>:
seq(LinearAlgebra:-Trace(T.M^n),n=0..100); # Robert Israel, Jan 28 2016

Join[{a=0,b=1},Table[c=2*b-4*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011 *)
LinearRecurrence[{2, -4}, {0, 1}, 40] (* Vincenzo Librandi, Jan 29 2016 *)
Table[2^(n-2)*((-1)^Quotient[n-1,3]+(-1)^Quotient[n,3]), {n,0,40}] (*Federico Provvedi,Apr 24 2022*)

/* lists powers of any quaternion */
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);} /* Stanislav Sykora, Jun 11 2012 */

my(x='x+O('x^30)); concat([0], Vec(x/(1-2*x+4*x^2))) \\ G. C. Greubel, Oct 22 2018

a(n) = 2^(n-1)*polchebyshev(n-1, 2, 1/2); \\ Michel Marcus, May 02 2022

[lucas_number1(n,2,4) for n in range(0, 39)] # Zerinvary Lajos, Apr 23 2009

G.f.: x/(1-2*x+4*x^2).
E.g.f.: exp(x)*sin(sqrt(3)*x)/sqrt(3).
a(n) = 2*a(n-1) - 4*a(n-2), a(0)=0, a(1)=1.
a(n) = ((1+i*sqrt(3))^n - (1-i*sqrt(3))^n)/(2*i*sqrt(3)).
a(n) = Im( (1+i*sqrt(3))^n/sqrt(3) ).
a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k+1)*(-3)^k.
From Paul Curtz, Oct 04 2009: (Start)
a(n) = a(n-1) + a(n-2) + 2*a(n-3).
a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3).
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4). (End)
E.g.f.: exp(x)*sin(sqrt(3)*x)/sqrt(3) = G(0)*x^2 where G(k)= 1 + (3*k+2)/(2*x - 32*x^5/(16*x^4 - 3*(k+1)*(3*k+2)*(3*k+4)*(3*k+5)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jul 26 2012
G.f.: x/(1-2*x+4*x^2) = 2*x^2*G(0) where G(k)= 1 + 1/(2*x - 32*x^5/(16*x^4 - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jul 27 2012
a(n) = -2^(n-1)*Product_{k=1..n}(1 + 2*cos(k*Pi/n)) for n >= 1. - Peter Luschny, Nov 28 2019
a(n) = 2^(n-1) * U(n-1, 1/2), where U(n, x) is the Chebyshev polynomial of the second kind. - Federico Provvedi, Apr 24 2022

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

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

A096252 Array read by rows, starting with n=0: row n lists A057077(n+1)8^(n+1)/2, A057077(n+2)8^(n+1)/2, A057077(n+1)*8^(n+1).

Original entry on oeis.org

4, -4, 8, -32, -32, -64, -256, 256, -512, 2048, 2048, 4096, 16384, -16384, 32768, -131072, -131072, -262144, -1048576, 1048576, -2097152, 8388608, 8388608, 16777216, 67108864, -67108864, 134217728, -536870912, -536870912, -1073741824

Offset: 0

Creighton Dement, Jul 31 2004

a(n) = ves( ('i + 'ii' + 'ij' + 'ik')^n ) a(n) = ves( ('j + 'jj' + 'ji' + 'jk')^n ) a(n) = ves( ('k + 'kk' + 'ki' + 'kj')^n ).
The elements x = 'i + 'ii' + 'ij' + 'ik'; y = 'j + 'jj' + 'ji' + 'jk'; and z = 'k + 'kk' + 'ki' + 'kj' are elements of the ring generated from the quaternion factor space Q X Q / {(1,1), (-1,-1)}. Each is represented by a gray shaded area of "Floret's cube". The elements x/2, y/2, z/2 are members of a group, itself a subset of the real algebra generated from Q X Q / {(1,1), (-1,-1)}, which is isomorphic to Q X C_3 (order 24).
This sequence is the term-wise sum of three sequences: a(n) = ves(x^n) = jes(x^n) + les(x^n) + tes(x^n), where jes(x^n)=(1, -6, 8, -24, 16, 0, -64, 384, -512, 1536, -1024, 0, 4096, -24576, 32768, -98304, ...), les(x^n)=(3, 0, 0, 0, -48, 0 -192, 0, 0, 0, 3072, 0, 12288, 0, 0, 0, ...), tes(x^n)=(0, 2, 0, -8, 0, -64, 0, -128, 0, 512, 0, 4096, 0, 8192, 0, -32768, ...). Concerning "les"- notice that if (..., s, 0, 0, 0, t, ...), then t = -16s and if (..., s, 0, t, ...), then t = 4s.

Danny Rorabaugh, Table of n, a(n) for n = 0..1000
C. Dement, The Math Forum.
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-16).

Cf. A048473, A094015.

CoefficientList[Series[4(1-x-2x^2-4x^3)/(1-4x^2+16x^4),{x,0,40}],x] (* or *) LinearRecurrence[ {0,4,0,-16},{4,-4,8,-32},40] (* Harvey P. Dale, Feb 15 2024 *)

[(-1)^(floor((floor(n/3)+((n%3)%2)+1)/2)) * 8^(floor(n/3)+1) / 2^(((n+1)^2)%3) for n in range(30)]
# Danny Rorabaugh, May 13 2016

a(n)= 4*a(n-2)-16*a(n-4). G.f.: 4*(1-x-2*x^2-4*x^3)/(1-4*x^2+16*x^4). - R. J. Mathar, Nov 26 2008
a(n) = (-1)^(floor((floor(n/3)+((n mod 3) mod 2)+1)/2)) * 8^(floor(n/3)+1) / 2^(((n+1)^2) mod 3). - Danny Rorabaugh, May 13 2016
a(n) = 4*(-1)^floor((n+1)/2)*A138230(n). - R. J. Mathar, May 21 2019

Edited with clearer definition by Omar E. Pol, Dec 29 2008

A202023 Triangle T(n,k), read by rows, given by (1, 0, 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.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 1, 6, 1, 0, 0, 1, 10, 5, 0, 0, 0, 1, 15, 15, 1, 0, 0, 0, 1, 21, 35, 7, 0, 0, 0, 0, 1, 28, 70, 28, 1, 0, 0, 0, 0, 1, 36, 126, 84, 9, 0, 0, 0, 0, 0, 1, 45, 210, 210, 45, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 10 2011

Riordan array (1/(1-x), x^2/(1-x)^2).
A skewed version of triangular array A085478.
Mirror image of triangle in A098158.
Sum_{k, 0<=k<=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 = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(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.
From Gus Wiseman, Jul 08 2025: (Start)
After the first row this is also the number of subsets of {1..n-1} with k maximal runs (sequences of consecutive elements increasing by 1) for k = 0..n. For example, row n = 5 counts the following subsets:
  {}  {1}        {1,3}    .  .  .
      {2}        {1,4}
      {3}        {2,4}
      {4}        {1,2,4}
      {1,2}      {1,3,4}
      {2,3}
      {3,4}
      {1,2,3}
      {2,3,4}
      {1,2,3,4}
Requiring n-1 gives A202064.
For anti-runs instead of runs we have A384893.
(End)

			Triangle begins :
1
1, 0
1, 1, 0
1, 3, 0, 0
1, 6, 1, 0, 0
1, 10, 5, 0, 0, 0
1, 15, 15, 1, 0, 0, 0
1, 21, 35, 7, 0, 0, 0, 0
1, 28, 70, 28, 1, 0, 0, 0, 0

Column k = 1 is A000217.
Column k = 2 is A000332.
Row sums are A011782 (or A000079 shifted right).
Removing all zeros gives A034839 (requiring n-1 A034867).
Last nonzero term in each row appears to be A093178, requiring n-1 A124625.
Reversing rows gives A098158, without zeros A109446.
Without the k = 0 column we get A210039.
Row maxima appear to be A214282.
A116674 counts strict partitions by number of maximal runs, for anti-runs A384905.
A268193 counts integer partitions by number of maximal runs, for anti-runs A384881.
Cf. A000045, A000071, A007318, A010027, A053538, A084938, A202064, A208342, A210034, A384175, A384893.

Table[Length[Select[Subsets[Range[n-1]],Length[Split[#,#2==#1+1&]]==k&]],{n,0,10},{k,0,n}] (* Gus Wiseman, Jul 08 2025 *)

T(n,k) = binomial(n,2k).
G.f.: (1-x)/((1-x)^2-y*x^2).
T(n,k)= Sum_{j, j>=0} T(n-1-j,k-1)*j with T(n,0)=1 and T(n,k)= 0 if k<0 or if nT(n,k) = 2*T(n-1,k) + T(n-2,k-1) - T(n-2,k) for n>1, T(0,0) = T(1,0) = 1, T(1,1) = 0, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Nov 10 2013

A157240 a(n) = A128018(n) + 1.

Original entry on oeis.org

2, -1, -7, -7, 17, 65, 65, -127, -511, -511, 1025, 4097, 4097, -8191, -32767, -32767, 65537, 262145, 262145, -524287, -2097151, -2097151, 4194305, 16777217, 16777217, -33554431, -134217727, -134217727, 268435457, 1073741825

Offset: 0

Creighton Dement, Feb 25 2009

Generating floretion is Y = .5('i + 'j + 'k + i' + j' + k') + ee. ("tes"). Note: A current conjecture is that if X is a floretion for which 4*tes(X^n) is an integer for all n, then X+sigma(X) also has this property. "sigma" is the uniquely defined projection operator which "flips the arrows" of a floretion (i.e. sigma('i) = i', sigma('j) = j', etc.). Taking X = .5('i + 'j + 'k + ee), then tesseq(X) = [ -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, ...] is an integer sequence, thus by the conjecture 4*tes(Y^n) = 4*tes((X+sigma)^n) should also be an integer sequence for all n.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-6,4).

Cf. A128018, A157241.

LinearRecurrence[{3,-6,4},{2,-1,-7},40] (* Harvey P. Dale, May 30 2021 *)

Vec((2 - 7*x + 8*x^2) / ((1 - x)*(1 - 2*x + 4*x^2)) + O(x^35)) \\ Colin Barker, May 22 2019

G.f.: (2-7*x+8*x^2) / ((1-x)*(4*x^2-2*x+1)).
From Colin Barker, May 22 2019: (Start)
a(n) = (2 + (1-i*sqrt(3))^(1+n) + (1+i*sqrt(3))^(1+n)) / 2 where i=sqrt(-1).
a(n) = 3*a(n-1) - 6*a(n-2) + 4*a(n-3) for n>2.
(End)
a(n) = A138230(n+1)+1. - R. J. Mathar, Sep 11 2019

A181738 T(n, k) is the coefficient of x^k of the polynomial p(n) which is defined as the scalar part of P(n) = Q(x+1, 1, 1, 1) * P(n-1) for n > 0 and P(0) = Q(1, 0, 0, 0) where Q(a, b, c, d) is a quaternion, triangle read by rows.

Original entry on oeis.org

1, 1, 1, -2, 2, 1, -8, -6, 3, 1, -8, -32, -12, 4, 1, 16, -40, -80, -20, 5, 1, 64, 96, -120, -160, -30, 6, 1, 64, 448, 336, -280, -280, -42, 7, 1, -128, 512, 1792, 896, -560, -448, -56, 8, 1, -512, -1152, 2304, 5376, 2016, -1008, -672, -72, 9, 1, -512, -5120, -5760, 7680, 13440, 4032, -1680, -960, -90, 10, 1

Offset: 0

Roger L. Bagula

The symbol '*' in the name refers to the noncommutative multiplication in Hamilton's division algebra. Traditionally Q(a, b, c, d) is written a + b*i + c*j + d*k.

			The list of polynomials starts 1, 1 + x, -2 + 2*x + x^2, -8 - 6*x + 3*x^2 + x^3, ... and the list of coefficients of the polynomials starts:
{   1},
{   1,     1},
{  -2,     2,     1},
{  -8,    -6,     3,    1},
{  -8,   -32,   -12,    4,     1},
{  16,   -40,   -80,  -20,     5,     1},
{  64,    96,  -120, -160,   -30,     6,     1},
{  64,   448,   336, -280,  -280,   -42,     7,    1},
{-128,   512,  1792,  896,  -560,  -448,   -56,    8,   1},
{-512, -1152,  2304, 5376,  2016, -1008,  -672,  -72,   9,  1},
{-512, -5120, -5760, 7680, 13440,  4032, -1680, -960, -90, 10, 1}.

Peter Luschny, Rows 0..45, flattened
Wikipedia, Quaternion

Cf. T(n,0) = A138230, A213421 (row sums).

Needs["Quaternions`"]
P[x_, 0 ] := Quaternion[1, 0, 0, 0];
P[x_, n_] := P[x, n] = Quaternion[x + 1, 1, 1, 1] ** P[x, n - 1];
Table[CoefficientList[P[x, n][[1]], x], {n, 0, 10}] // Flatten

R. = QQ[]
K = R.fraction_field()
H. = QuaternionAlgebra(K, -1, -1)
def Q(a, b, c, d): return H(a + b*i + c*j + d*k)
@cached_function
def P(n):
    return Q(x+1,1,1,1)*P(n-1) if n > 0 else Q(1,0,0,0)
def p(n): return P(n)[0].numerator().list()
flatten([p(n) for n in (0..10)]) # Kudos to William Stein, Peter Luschny, Sep 14 2018

Edited by Peter Luschny, Sep 14 2018

A221131 Table, T, read by antidiagonals where T(-j,k) = ((1+sqrt(j))^k + (1-sqrt(j))^k)/2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -2, 1, 1, 1, -2, -5, -4, 1, 1, 1, -3, -8, -7, -4, 1, 1, 1, -4, -11, -8, 1, 0, 1, 1, 1, -5, -14, -7, 16, 23, 8, 1, 1, 1, -6, -17, -4, 41, 64, 43, 16, 1, 1, 1, -7, -20, 1, 76, 117, 64, 17, 16, 1, 1, 1, -8, -23, 8, 121, 176, 29, -128, -95, 0, 1

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com) and Robert G. Wilson v, Jan 02 2013

.j\k.........0..1...2....3...4....5....6......7.......8......9......10
.0: A000012..1..1...1....1...1....1....1......1.......1......1.......1
-1: A146559..1..1...0...-2..-4...-4....0......8......16.....16.......0
-2: A087455..1..1..-1...-5..-7....1...23.....43......17....-95....-241
-3: A138230..1..1..-2...-8..-8...16...64.....64....-128...-512....-512
-4: A006495..1..1..-3..-11..-7...41..117.....29....-527..-1199.....237
-5: A138229..1..1..-4..-14..-4...76..176...-104...-1264..-1904....3776
-6: A090592..1..1..-5..-17...1..121..235...-377...-2399..-2159...12475
-7: A090590..1..1..-6..-20...8..176..288...-832...-3968..-1280...29184
-8: A025172..1..1..-7..-23..17..241..329..-1511...-5983...1633...57113
-9: A120743..1..1..-8..-26..28..316..352..-2456...-8432...7696...99712
-10: ........1..1..-9..-29..41..401..351..-3709..-11279..18241..160551

Cf. A084097,
Rows: A000012, A146559, A087455, A138230, A006495, A138229, A090592, A090590, A025172, A120743.
Columns: A000012, A000012, A024000, 1-3n, A028884, A134593.

T[j_, k_] := Expand[((1 + Sqrt[j])^k + (1 - Sqrt[j])^k)/2]; Table[ T[ -j + k, k], {j, 0, 11}, {k, 0, j}] // Flatten

Showing 1-10 of 13 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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Magma

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PARI

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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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Maple

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A088138 Generalized Gaussian Fibonacci integers.

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A124182 A skewed version of triangular array A081277.

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

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A096252 Array read by rows, starting with n=0: row n lists A057077(n+1)*8^(n+1)/2, A057077(n+2)*8^(n+1)/2, A057077(n+1)*8^(n+1).

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

A096252 Array read by rows, starting with n=0: row n lists A057077(n+1)8^(n+1)/2, A057077(n+2)8^(n+1)/2, A057077(n+1)*8^(n+1).