A084099 -id:A084099 - cp's OEIS frontend

A007877 Period 4 zigzag sequence: repeat [0,1,2,1].

0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0

Offset: 0

Christopher Lam Cham Kee (Topher(AT)CyberDude.Com)

Euler transform of finite sequence [2,-2,0,1]. - Michael Somos, Sep 17 2004
This is the r = 2 member in the r-family of sequences S_r(n) defined in A092184 where more information can be found.
a(n+1) is the transform of sqrt(1+2x)/sqrt(1-2x) (A063886) under the Chebyshev transformation A(x) -> (1/(1 + x^2))A(x/(1 + x^2)). See also A084099. - Paul Barry, Oct 12 2004
Multiplicative with a(2) = 2, a(2^e) = 0 if e >= 2, a(p^e) = 1 otherwise. - David W. Wilson, Jun 12 2005
The e.g.f. of 1, 2, 1, 0, 1, 2, 1, 0, ... (shifted left, offset zero) is exp(x) + sin(x).
Binomial transform is A000749(n+2). - Wesley Ivan Hurt, Dec 30 2015
Decimal expansion of 11/909. - David A. Corneth, Dec 12 2016
Ternary expansion of 1/5. - J. Conrad, Aug 14 2017

P. Liu, Efficient recognition of integer sequences, Master's Essay, University of Waterloo, Dec. 1994. (Annotated scanned copy)
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (1,-1,1).

Cf. A000749, A021913, A049310, A056594, A063886, A084099, A092184, A181878.

Period k zigzag sequences: A000035 (k=2), this sequence (k=4), A260686 (k=6), A266313 (k=8), A271751 (k=10), A271832 (k=12), A279313 (k=14), A279319 (k=16), A158289 (k=18).

&cat [[0,1,2,1]^^25]; // Vincenzo Librandi, Dec 27 2015

A007877:=n->sqrt(n^2 mod 8); seq(A007877(n), n=0..100); # Wesley Ivan Hurt, Jan 01 2014

f[n_] := Mod[n, 4] - Mod[n^3, 4] + Mod[n^2, 4] (* Or *)
f[n_] := Mod[n, 2] + 2 Floor[Mod[n + 1, 4]/3] (* Or *)
f[n_] := Switch[Mod[n, 4], 0, 0, 1, 1, 2, 2, 3, 1]; Array[f, 105, 0] (* Robert G. Wilson v, Aug 08 2011 *)
Table[Sqrt[Mod[n^2,8]], {n,0,100}] (* Wesley Ivan Hurt, Jan 01 2014 *)
LinearRecurrence[{1, -1, 1}, {0, 1, 2}, 80] (* Vincenzo Librandi, Dec 27 2015 *)
PadRight[{},100,{0,1,2,1}] (* Harvey P. Dale, Oct 24 2023 *)

a(n)=[0,1,2,1][1+n%4] \\ Jaume Oliver Lafont, Mar 27 2009

concat(0, Vec(x*(1+x)/(1-x+x^2-x^3) + O(x^100))) \\ Altug Alkan, Dec 29 2015

def A007877(n): return (0,1,2,1)[n&3] # Chai Wah Wu, Jan 26 2023

Multiplicative with a(p^e) = 2 if p = 2 and e = 0; 0 if p = 2 and e > 0; 1 if p > 2. - David W. Wilson, Aug 01 2001
a(n) = -Sum_{k=0..n} (-1)^C(k+2, 2) (Offset -1). - Paul Barry, Jul 07 2003
a(n) = 1 - cos(n*Pi/2); a(n) = a(n-1) - a(n-2) + a(n-3) for n>2. - Lee Reeves (leereeves(AT)fastmail.fm), May 10 2004
a(n) = -a(n-2) + 2, n >= 2, a(0) = 0, a(1) = 1.
G.f.: x*(1+x)/((1-x)*(1+x^2)) = x*(1+x)/(1-x+x^2-x^3).
a(n) = 1 - T(n, 0) = 1 - A056594(n) with Chebyshev's polynomials T(n, x) of the first kind. Note that T(n, 0) = S(n, 0).
a(n) = b(n) + b(n-1), n >= 1, with b(n) := A021913(n+1) the partial sums of S(n,0) = U(n,0) = A056594(n) (Chebyshev's polynomials evaluated at x=0).
a(n) = 1 + (1/2){(-1)^[(n-1)/2] - (-1)^[n/2]}. - Ralf Stephan, Jun 09 2005
Non-reduced g.f.: x*(1+x)^2/(1-x^4). - Jaume Oliver Lafont, Mar 27 2009
a(n+1) = (S(n, sqrt(2)))^2, n >= 0, with the Chebyshev S-polynomials A049310. See the W. Lang link under A181878. - Wolfdieter Lang, Dec 15 2010
Dirichlet g.f. (1 + 1/2^s - 2/4^s)*zeta(s). - R. J. Mathar, Feb 24 2011
a(n) = (n mod 4) - (n^3 mod 4) + (n^2 mod 4). - Gary Detlefs, Apr 17 2011
a(n) = (n mod 2) + 2*floor(((n+1) mod 4)/3). - Gary Detlefs, Jul 19 2011
a(n) = sqrt(n^2 mod 8). - Wesley Ivan Hurt, Jan 01 2014
a(n) = (n AND 4*k+2)-(n AND 4*k+1) + 2*floor(((n+2) mod 4)/3), for any k. - Gary Detlefs, Jun 08 2014
a(n) = Sum_{i=1..n} (-1)^floor((i-1)/2). - Wesley Ivan Hurt, Dec 26 2015
a(n) = a(n-4) for n >= 4. - Wesley Ivan Hurt, Sep 07 2022
a(n) = n - 2*floor(n/4) - 2*floor((n+1)/4). - Ridouane Oudra, Jan 22 2024
E.g.f.: exp(x) - cos(x). - Stefano Spezia, Aug 04 2025

Chebyshev comments from Wolfdieter Lang, Sep 10 2004

A176742 Expansion of (1 - x^2) / (1 + x^2) in powers of x.

Original entry on oeis.org

1, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2

Offset: 0

Wolfdieter Lang, Oct 15 2010

Difference sequence of A057077.
Sequence of determinants of matrices for some bipartite graphs, called Tz(n). The graph Tz(4) appears in the logo for the beer called Tannenzäpfle (small fir cone), brewed by Badische Staatsbrauerei Rothaus, Germany, hence the name Tz. See the link for this logo with Tz(4).
The vertex-vertex matrix for these bipartite graphs will also be called Tz(n) (without leading to confusion).
General proof by expanding the determinant a(n) = determinant(Tz(n)) along the first column yielding b(n-1)-b(n-2), with b(n-1) the A_{1,1} minor of the matrix Tz(n), and deriving a recurrence for the b(n), namely b(n) = -b(n-2) with inputs b(0) = 1 = b(1). This gives b(n) = A057077(n), n>=0.

			G.f. = 1 - 2*x^2 + 2*x^4 - 2*x^6 + 2*x^8 - 2*x^10 + 2*x^12 - 2*x^14 + 2*x^16 + ...
The bipartite graphs Tz(n) (n>=1) look like |, |X|, |XX|, |XXX|, ... For n>=2 the lines have to be connected to give the 2*n nodes and 2*n edges. The n=1 graph Tz(1) has 2*1=2 nodes and only one edge.
n=1: determinant((1))=1; n=2: determinant(Matrix([[1,1],[1,1]]))=0; n=3: determinant(Matrix([[1,1,0],[1,0,1],[0,1,1]]))=-2; n=4: determinant(Tz(4))=0; etc.

Wolfdieter Lang, Some Tz(n) graphs and matrices.
Rothaus Tannenzäpfle, Tz(4) graph on the Tannenzäpfle logo
Index entries for linear recurrences with constant coefficients, signature (0,-1).

Cf. A084099, A057077.

a := n -> 2^signum(n)*(-1)^iquo(n+1,2)*modp(n+1,2);
seq(a(n), n=0..100); # Peter Luschny, Jun 22 2014

Join[{1}, Table[{0, -2, 0, 2}, {26}]] // Flatten (* Jean-François Alcover, Jun 21 2013 *)
a[ n_] := - Boole[n == 0] + {0, -2, 0, 2}[[Mod[ n, 4, 1]]]; (* Michael Somos, May 05 2015 *)
PadRight[{1},120,{2,0,-2,0}] (* Harvey P. Dale, Apr 13 2019 *)

{a(n) = - (n == 0) + [2, 0, -2, 0][n%4 + 1]}; /* Michael Somos, Mar 21 2011 */

def A176742(n): return (2,0,-2,0)[n&3] if n else 1 # Chai Wah Wu, Apr 22 2025

Euler transform of length 4 sequence [0, -2, 0, 1]. - Michael Somos, Mar 21 2011
Moebius transform is length 4 sequence [0, -2, 0, 4]. - Michael Somos, Mar 22 2011
a(n) = a(-n) for all n in Z. a(n) = c_4(n) if n>1, where c_k(n) is Ramanujan's sum. - Michael Somos, Mar 21 2011
a(n-1) := determinant(Tz(n)), n>=1. The rows of the matrix Tz(4) are [[1, 1, 0, 0], [1, 0, 1, 0], [0, 1, 0, 1], [0, 0, 1, 1]]. Tz(1)=(1), and Tz(2) has rows [[1, 1], [1, 1]]. The matrix for the generalization Tz(n) has rows [[1,1,0,...,0], [1,0,1,0,...,0], [0,1,0,1,0,...,0], ..., [0,...,0,1,0,1], [0,...,0,1,1].
a(0)=1, a(2*k-1)= 0, a(4*k) = +2, a(4*k-2) = -2, k>=1.
O.g.f.: (1-x^2)/(1+x^2).
a(n) = A057077(n) - A057077(n-1), n>=1. a(0)=1.
Dirichlet g.f. sum_{n>=1} a(n)/n^s = zeta(s)*(4^(1-s)-2^(1-s)). - R. J. Mathar, Apr 11 2011
a(n) = (((n+1) mod 2)+((n+2+sign(n)) mod 2))*(-1)^ceiling(n/2). - Wesley Ivan Hurt, Jun 20 2014

A128908 Riordan array (1, x/(1-x)^2).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 4, 10, 6, 1, 0, 5, 20, 21, 8, 1, 0, 6, 35, 56, 36, 10, 1, 0, 7, 56, 126, 120, 55, 12, 1, 0, 8, 84, 252, 330, 220, 78, 14, 1, 0, 9, 120, 462, 792, 715, 364, 105, 16, 1, 0, 10, 165, 792, 1716, 2002, 1365, 560, 136, 18, 1

Offset: 0

Philippe Deléham, Apr 22 2007

Triangle T(n,k), 0 <= k <= n, read by rows given by [0,2,-1/2,1/2,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
Row sums give A088305. - Philippe Deléham, Nov 21 2007
Column k is C(n,2k-1) for k > 0. - Philippe Deléham, Jan 20 2012
From R. Bagula's comment in A053122 (cf. Damianou link p. 10), this array gives the coefficients (mod sign) of the characteristic polynomials for the Cartan matrix of the root system A_n. - Tom Copeland, Oct 11 2014
T is the convolution triangle of the positive integers (see A357368). - Peter Luschny, Oct 19 2022

			The triangle T(n,k) begins:
   n\k  0    1    2    3    4    5    6    7    8    9   10
   0:   1
   1:   0    1
   2:   0    2    1
   3:   0    3    4    1
   4:   0    4   10    6    1
   5:   0    5   20   21    8    1
   6:   0    6   35   56   36   10    1
   7:   0    7   56  126  120   55   12    1
   8:   0    8   84  252  330  220   78   14    1
   9:   0    9  120  462  792  715  364  105   16    1
  10:   0   10  165  792 1716 2002 1365  560  136   18    1
  ... reformatted by _Wolfdieter Lang_, Jul 31 2017
From _Peter Luschny_, Mar 06 2022: (Start)
The sequence can also be seen as a square array read by upwards antidiagonals.
   1, 1,   1,    1,    1,     1,     1,      1,      1, ...  A000012
   0, 2,   4,    6,    8,    10,    12,     14,     16, ...  A005843
   0, 3,  10,   21,   36,    55,    78,    105,    136, ...  A014105
   0, 4,  20,   56,  120,   220,   364,    560,    816, ...  A002492
   0, 5,  35,  126,  330,   715,  1365,   2380,   3876, ... (A053126)
   0, 6,  56,  252,  792,  2002,  4368,   8568,  15504, ... (A053127)
   0, 7,  84,  462, 1716,  5005, 12376,  27132,  54264, ... (A053128)
   0, 8, 120,  792, 3432, 11440, 31824,  77520, 170544, ... (A053129)
   0, 9, 165, 1287, 6435, 24310, 75582, 203490, 490314, ... (A053130)
    A27,A292, A389, A580,  A582, A1288, A10966, A10968, A165817       (End)

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv:1110.6620 [math.RT], 2014.

Cf. A002450, A007318, A034008, A053122, A078812, A084938, A088305.
Cf. Columns : A000007, A000027, A000292, A000389, A000580, A000582, A001288, A010966 ..(+2).. A011000, A017713 ..(+2).. A017763.
Cf. A000007, A001352, A008574, A054888, A084099, A084103, A163810, A357368.
Cf. A165817 (the main diagonal of the array).

# Computing the rows of the array representation:
S := proc(n,k) option remember;
if n = k then 1 elif k < 0 or k > n then 0 else
S(n-1, k-1) + 2*S(n-1, k) - S(n-2, k) fi end:
Arow := (n, len) -> seq(S(n+k-1, k-1), k = 0..len-1):
for n from 0 to 8 do Arow(n, 9) od; # Peter Luschny, Mar 06 2022
# Uses function PMatrix from A357368.
PMatrix(10, n -> n); # Peter Luschny, Oct 19 2022

With[{nmax = 10}, CoefficientList[CoefficientList[Series[(1 - x)^2/(1 - (2 + y)*x + x^2), {x, 0, nmax}, {y, 0, nmax}], x], y]] // Flatten (* G. C. Greubel, Nov 22 2017 *)

for(n=0,10, for(k=0,n, print1(if(n==0 && k==0, 1, if(k==0, 0, binomial(n+k-1,2*k-1))), ", "))) \\ G. C. Greubel, Nov 22 2017

from functools import cache
@cache
def A128908(n, k):
    if n == k: return 1
    if (k <= 0 or k > n): return 0
    return A128908(n-1, k-1) + 2*A128908(n-1, k) - A128908(n-2, k)
for n in range(10):
    print([A128908(n, k) for k in range(n+1)]) # Peter Luschny, Mar 07 2022

@cached_function
def T(k,n):
    if k==n: return 1
    if k==0: return 0
    return sum(i*T(k-1,n-i) for i in (1..n-k+1))
A128908 = lambda n,k: T(k,n)
for n in (0..10): print([A128908(n,k) for k in (0..n)]) # Peter Luschny, Mar 12 2016

T(n,0) = 0^n, T(n,k) = binomial(n+k-1, 2k-1) for k >= 1.
Sum_{k=0..n} T(n,k)*2^(n-k) = A002450(n) = (4^n-1)/3 for n>=1. - Philippe Deléham, Oct 19 2008
G.f.: (1-x)^2/(1-(2+y)*x+x^2). - Philippe Deléham, Jan 20 2012
Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A001352(n), (-1)^(n+1)*A054888(n+1), (-1)^n*A008574(n), (-1)^n*A084103(n), (-1)^n*A084099(n), A163810(n), A000007(n), A088305(n) for x = -6, -5, -4, -3, -2, -1, 0, 1 respectively. - Philippe Deléham, Jan 20 2012
Riordan array (1, x/(1-x)^2). - Philippe Deléham, Jan 20 2012

A084101 Expansion of (1+x)^2/((1-x)*(1+x^2)).

Original entry on oeis.org

1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1

Offset: 0

Paul Barry, May 15 2003

Partial sums of A084099. Inverse binomial transform of A000749 (without leading zeros).
From Klaus Brockhaus, May 31 2010: (Start)
Periodic sequence: Repeat 1, 3, 3, 1.
Interleaving of A010684 and A176040.
Continued fraction expansion of (7 + 5*sqrt(29))/26.
Decimal expansion of 121/909.
a(n) = A143432(n+3) + 1 = 2*A021913(n+1) + 1 = 2*A133872(n+3) + 1.
a(n) = A165207(n+1) - 1.
First differences of A047538.
Binomial transform of A084102. (End)
From Wolfdieter Lang, Feb 09 2012: (Start)
a(n) = A045572(n+1) (Modd 5) := A203571(A045572(n+1)), n >= 0.
For general Modd n (not to be confused with mod n) see a comment on A203571. The nonnegative members of the five residue classes Modd 5, called [m] for m=0,1,...,4, are shown in the array A090298 if there the last row is taken as class [0] after inclusion of 0.
(End)

			From _Wolfdieter Lang_, Feb 09 2012: (Start)
Modd 5 of nonnegative odd numbers restricted mod 5:
A045572: 1, 3, 7, 9, 11, 13, 17, 19, 21, 23, ...
Modd 5:  1, 3, 3, 1,  1,  3,  3,  1,  1,  3, ...
(End)

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

Cf. A084102.

Cf. A010684 (repeat 1, 3), A176040 (repeat 3, 1), A178593 (decimal expansion of (7+5*sqrt(29))/26), A143432 (expansion of (1+x^4)/((1-x)*(1+x^2))), A021913 (repeat 0, 0, 1, 1), A133872 (repeat 1, 1, 0, 0), A165207 (repeat 2, 2, 4, 4), A047538 (congruent to 0, 1, 4 or 7 mod 8), A084099 (expansion of (1+x)^2/(1+x^2)), A000749 (expansion of x^3/((1-x)^4-x^4)). - Klaus Brockhaus, May 31 2010

R:=PowerSeriesRing(Integers(), 100); Coefficients(R!( (1+x)^2/((1-x)*(1+x^2)) )); // G. C. Greubel, Feb 28 2019

CoefficientList[Series[(1+x)^2/((1-x)(1+x^2)),{x,0,110}],x] (* or *) PadRight[{},110,{1,3,3,1}] (* Harvey P. Dale, Nov 21 2012 *)

x='x+O('x^100); Vec((1+x)^2/((1-x)*(1+x^2))) \\ Altug Alkan, Dec 24 2015

((1+x)^2/((1-x)*(1+x^2))).series(x, 100).coefficients(x, sparse=False) # G. C. Greubel, Feb 28 2019

a(n) = binomial(3, n mod 4). - Paul Barry, May 25 2003
From Klaus Brockhaus, May 31 2010: (Start)
a(n) = a(n-4) for n > 3; a(0) = a(3) = 1, a(1) = a(2) = 3.
a(n) = (4 - (1+i)*i^n - (1-i)*(-i)^n)/2 where i = sqrt(-1). (End)
E.g.f.: 2*exp(x) + sin(x) - cos(x). - Arkadiusz Wesolowski, Nov 04 2017
a(n) = 2 - (-1)^(n*(n+1)/2). - Guenther Schrack, Feb 26 2019

A383987 Series expansion of the exponential generating function -tridend(-(1-exp(x))) where tridend(x) = (1 - 3x - sqrt(1+6x+x^2)) / (4*x) (A001003).

Original entry on oeis.org

0, 1, -5, 49, -725, 14401, -360005, 10863889, -384415925, 15612336481, -715930020005, 36592369889329, -2062911091119125, 127170577711282561, -8510569547826528005, 614491222512504748369, -47615614242877583230325, 3941408640018910366196641

Offset: 0

Michael De Vlieger, May 16 2025

Michael De Vlieger, Table of n, a(n) for n = 0..353
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28, Table 2, triassociative operad "Trias".

Cf. A002050, A006531, A084099, A101851, A114285, A225883, A383985, A383986, A383988, A383989, A383991.

Composition of A001003 with exp(x)-1.

nn = 17; f[x_] := (1 + 3*x - Sqrt[1 + 6*x + x^2])/(4*x); Range[0, nn]! * CoefficientList[Series[f[-(1 - Exp[x])], {x, 0, nn}], x]

A383989 Series expansion of the exponential generating function ff6^!(exp(x)-1) where ff6^!(x) = x * (1-3x-x^2+x^3) / (1+3x+x^2-x^3).

Original entry on oeis.org

0, 1, -11, 61, -467, 4381, -49091, 643021, -9615827, 161844541, -3026079971, 62243374381, -1396619164787, 33949401567901, -888725861445251, 24926889744928141, -745755560487363347, 23705772035082494461, -797875590555470224931, 28346366547928396344301

Offset: 0

Michael De Vlieger, May 16 2025

Michael De Vlieger, Table of n, a(n) for n = 0..407
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28, Table 2, operad "FF6".

Cf. A002050, A006531, A084099, A101851, A114285, A225883, A383985, A383986, A383987, A383988, A383995.

nn = 19; f[x_] := x*(1 - 3*x - x^2 + x^3)/(1 + 3*x + x^2 - x^3);
Range[0, nn]! * CoefficientList[Series[f[-(1 - Exp[x])], {x, 0, nn}], x]

A383994 Series expansion of the exponential generating function exp(wnp^!(x)) - 1 where wnp^!(x) = log(1+x) - x^2/(1+x).

Original entry on oeis.org

0, 1, -2, 0, 12, -60, 240, -840, 1680, 15120, -332640, 4656960, -59209920, 735134400, -9098369280, 112345833600, -1365274310400, 15746578848000, -155630893017600, 762963647846400, 22567767443020800, -1126188650069683200, 35900904478389350400

Offset: 0

Michael De Vlieger, May 16 2025

sign

The series wnp^!(x) is the inverse for the substitution of the series wnp(x) (corresponding to A048172), given by the suspension of the Koszul dual of the WithoutNPosets operad. - Bérénice Delcroix-Oger, May 28 2025

Michael De Vlieger, Table of n, a(n) for n = 0..451
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 32, Table 3, operad without n posets "WNP".

Cf. A003725, A084099, A097388, A111884, A112242, A177885, A318215, A383990, A383991, A383992, A383993, A383995.

nn = 22; f[x_] := Exp[x] - 1;
Range[0, nn]! * CoefficientList[Series[f[Log[1 + x] - x^2/(1 + x)], {x, 0, nn}], x]

A105343 Elements of even index in the sequence gives A005893, points on surface of tetrahedron: 2n^2 + 2 for n > 1.

Original entry on oeis.org

1, 3, 4, 7, 10, 15, 20, 27, 34, 43, 52, 63, 74, 87, 100, 115, 130, 147, 164, 183, 202, 223, 244, 267, 290, 315, 340, 367, 394, 423, 452, 483, 514, 547, 580, 615, 650, 687, 724, 763, 802, 843, 884, 927, 970, 1015, 1060, 1107, 1154, 1203, 1252, 1303, 1354, 1407

Offset: 0

Creighton Dement, Apr 30 2005

Floretion Algebra Multiplication Program, FAMP Code: 2jesforrokseq[E*F*sig(E)] with E = + .5i' + .5j' + .5'ki' + .5'kj', F the sum of all floretion basis vectors and "sig" the swap-operator. RokType: Y[15] = Y[15] + Math.signum(Y[15])*p (internal program code)
May be seen as the jesforrok-transform of the zero-sequence (A000004) with respect to the floretion given in the program code.
Identical to A267459(n+1) for n > 0. - Guenther Schrack, Jun 01 2018

			G.f. = 1 + 3*x + 4*x^2 + 7*x^3 + 10*x^4 + 15*x^5 + 20*x^6 + 27*x^7 + ... - _Michael Somos_, Jun 26 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A005893, A084099, A267459.

[1] cat [(2*n^2 + 9 - (-1)^n) div 4: n in [1..60]]; // Vincenzo Librandi, Oct 10 2011

Join[{1}, LinearRecurrence[{2, 0, -2, 1}, {3, 4, 7, 10}, 60]] (* Jean-François Alcover, Nov 13 2017 *)

{a(n) = if( n<1, n==0, (2*n^2 + 10)\4)}; /* Michael Somos, Jun 26 2018 */

G.f.: (1 + x - 2*x^2 + x^3 + x^4)/((x+1)*(1-x)^3); a(n+2) - 2*a(n+1) + a(n) = (-1)^(n+1)*A084099(n).
a(n) = (1/4)*(2*n^2 + 9 - (-1)^n ), n>1. - Ralf Stephan, Jun 01 2007
Sum_{n>=0} 1/a(n) = 3/4 + tanh(sqrt(5)*Pi/2)*Pi/(2*sqrt(5)) + coth(Pi)*Pi/4. - Amiram Eldar, Sep 16 2022

A383985 Series expansion of the exponential generating function LambertW(1-exp(x)), see A000169.

Original entry on oeis.org

0, 1, -1, 4, -23, 181, -1812, 22037, -315569, 5201602, -97009833, 2019669961, -46432870222, 1168383075471, -31939474693297, 942565598033196, -29866348653695203, 1011335905644178273, -36446897413531401020, 1392821757824071815641, -56259101478392975833333

Offset: 0

Michael De Vlieger, May 16 2025

Michael De Vlieger, Table of n, a(n) for n = 0..200
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28, Table 2, permutative commutative operad "Perm/Com2".

Cf. A002050, A006531, A084099, A101851, A114285, A177885, A225883, A383986, A383987, A383988, A383989.

Composition of A000169 with signs and 1-exp(x).

nn = 20; f[x_] := -Sum[k^(k - 1)*(1 - Exp[x])^k/k!, {k, nn}];
Range[0, nn]! * CoefficientList[Series[f[x], {x, 0, nn}], x]

A383986 Expansion of the exponential generating function sqrt(4exp(x) - exp(2x) - 2) - 1.

Original entry on oeis.org

0, 1, -1, 1, -13, 61, -601, 5881, -73333, 1021861, -16334401, 290146561, -5707536253, 122821558861, -2873553719401, 72586328036041, -1969306486088773, 57106504958139061, -1762735601974347601, 57705363524117482321, -1996916624448159410893

Offset: 0

Michael De Vlieger, May 16 2025

Michael De Vlieger, Table of n, a(n) for n = 0..409
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28, Table 2, operad "NAC_2".

Cf. A002050, A006531, A084099, A101851, A114285, A182037, A225883, A383985, A383987, A383988, A383989.

nn = 20; f[x_] := -1 + Sqrt[1 + 2 x - x^2];
Range[0, nn]! * CoefficientList[Series[f[-(1 - Exp[x])], {x, 0, nn}], x]

cp's OEIS Frontend

A007877 Period 4 zigzag sequence: repeat [0,1,2,1].

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A176742 Expansion of (1 - x^2) / (1 + x^2) in powers of x.

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A128908 Riordan array (1, x/(1-x)^2).

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A084101 Expansion of (1+x)^2/((1-x)*(1+x^2)).

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A383987 Series expansion of the exponential generating function -tridend(-(1-exp(x))) where tridend(x) = (1 - 3*x - sqrt(1+6*x+x^2)) / (4*x) (A001003).

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A383989 Series expansion of the exponential generating function ff6^!(exp(x)-1) where ff6^!(x) = x * (1-3*x-x^2+x^3) / (1+3*x+x^2-x^3).

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A383994 Series expansion of the exponential generating function exp(wnp^!(x)) - 1 where wnp^!(x) = log(1+x) - x^2/(1+x).

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A383987 Series expansion of the exponential generating function -tridend(-(1-exp(x))) where tridend(x) = (1 - 3x - sqrt(1+6x+x^2)) / (4*x) (A001003).

A383989 Series expansion of the exponential generating function ff6^!(exp(x)-1) where ff6^!(x) = x * (1-3x-x^2+x^3) / (1+3x+x^2-x^3).

A383986 Expansion of the exponential generating function sqrt(4exp(x) - exp(2x) - 2) - 1.