A154955 -id:A154955 - cp's OEIS frontend

A000035 Period 2: repeat [0, 1]; a(n) = n mod 2; parity of n.

0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0

Offset: 0

N. J. A. Sloane

Least significant bit of n, lsb(n).
Also decimal expansion of 1/99.
Also the binary expansion of 1/3. - Robert G. Wilson v, Sep 01 2015
a(n) = A134451(n) mod 2. - Reinhard Zumkeller, Oct 27 2007 [Corrected by Jianing Song, Nov 22 2019]
Characteristic function of odd numbers: a(A005408(n)) = 1, a(A005843(n)) = 0. - Reinhard Zumkeller, Sep 29 2008
A102370(n) modulo 2. - Philippe Deléham, Apr 04 2009
Base b expansion of 1/(b^2-1) for any b >= 2 is 0.0101... (A005563 has b^2-1). - Rick L. Shepherd, Sep 27 2009
Let A be the Hessenberg n X n matrix defined by: A[1,j] = j mod 2, A[i,i] := 1, A[i,i-1] = -1, and A[i,j] = 0 otherwise. Then, for n >= 1, a(n) = (-1)^n*charpoly(A,1). - Milan Janjic, Jan 24 2010
From R. J. Mathar, Jul 15 2010: (Start)
The sequence is the principal Dirichlet character of the reduced residue system mod 2 or mod 4 or mod 8 or mod 16 ...
Associated Dirichlet L-functions are for example L(2,chi) = Sum_{n>=1} a(n)/n^2 == A111003,
or L(3,chi) = Sum_{n>=1} a(n)/n^3 = 1.05179979... = 7*A002117/8,
or L(4,chi) = Sum_{n>=1} a(n)/n^4 = 1.014678... = A092425/96. (End)
Also parity of the nonnegative integers A001477. - Omar E. Pol, Jan 17 2012
a(n) = (4/n), where (k/n) is the Kronecker symbol. See the Eric Weisstein link. - Wolfdieter Lang, May 28 2013
Also the inverse binomial transform of A131577. - Paul Curtz, Nov 16 2016 [an observation forwarded by Jean-François Alcover]
The emanation sequence for the globe category. That is take the globe category, take the corresponding polynomial comonad, consider its carrier polynomial as a generating function, and take the corresponding sequence. - David Spivak, Sep 25 2020
For n > 0, a(n) is the alternating sum of the product of n increasing and n decreasing odd factors. For example, a(4) = 1*7 - 3*5 + 5*3 - 7*1 and a(5) = 1*9 - 3*7 + 5*5 - 7*3 + 9*1. - Charlie Marion, Mar 24 2022

			G.f. = x + x^3 + x^5 + x^7 + x^9 + x^11 + x^13 + x^15 + ... - _Michael Somos_, Feb 20 2024

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David Wasserman, Table of n, a(n) for n = 0..1000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Dirichlet Series Generating Function
Eric Weisstein's World of Mathematics, Kronecker Symbol
A. K. Whitford, Binet's Formula Generalized, Fibonacci Quarterly, Vol. 15, No. 1, 1979, pp. 21, 24, 29
Index entries for "core" sequences
Index entries for sequences that are fixed points of mappings
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,1).

Ones complement of A059841.
Cf. A053644 for most significant bit.
This is Guy Steele's sequence GS(1, 2) (see A135416).
Period k zigzag sequences: this sequence (k=2), A007877 (k=4), A260686 (k=6), A266313 (k=8), A271751 (k=10), A271832 (k=12), A279313 (k=14), A279319 (k=16), A158289 (k=18).
Cf. A154955 (Mobius transform), A131577 (binomial transform).
Cf. A111003 (Dgf at s=2), A233091 (Dgf at s=3), A300707 (Dgf at s=4).
Parity of A005811.

a000035 n = n `mod` 2  -- James Spahlinger, Oct 08 2012

a000035_list = cycle [0,1]  -- Reinhard Zumkeller, Jan 06 2012

A000035 := n->n mod 2;
[ seq(i mod 2, i=0..100) ];

PadLeft[{},110,{0,1}] (* Harvey P. Dale, Sep 25 2011 *)

A000035(n):=mod(n,2)$
makelist(A000035(n),n,0,30); /* Martin Ettl, Nov 12 2012 */

PARI
```
a(n)=n%2;
```

a(n)=direuler(p=1,100,if(p==2,1,1/(1-X)))[n] /* Ralf Stephan, Mar 27 2015 */

def A000035(n): return n & 1 # Chai Wah Wu, May 25 2022

(define (A000035 n) (mod n 2)) ;; For R6RS. Use modulo in older Schemes like MIT/GNU Scheme. - Antti Karttunen, Mar 21 2017

a(n) = (1 - (-1)^n)/2.
a(n) = n mod 2.
a(n) = 1 - a(n-1).
Multiplicative with a(p^e) = p mod 2. - David W. Wilson, Aug 01 2001
G.f.: x/(1-x^2). E.g.f.: sinh(x). - Paul Barry, Mar 11 2003
a(n) = (A000051(n) - A014551(n))/2. - Mario Catalani (mario.catalani(AT)unito.it), Aug 30 2003
a(n) = ceiling((-2)^(-n-1)). - Reinhard Zumkeller, Apr 19 2005
Dirichlet g.f.: (1-1/2^s)*zeta(s). - R. J. Mathar, Mar 04 2011
a(n) = ceiling(n/2) - floor(n/2). - Arkadiusz Wesolowski, Sep 16 2012
a(n) = ceiling( cos(Pi*(n-1))/2 ). - Wesley Ivan Hurt, Jun 16 2013
a(n) = floor((n-1)/2) - floor((n-2)/2). - Mikael Aaltonen, Feb 26 2015
Dirichlet g.f.: L(chi(2),s) with chi(2) the principal Dirichlet character modulo 2. - Ralf Stephan, Mar 27 2015
a(n) = 0^^n = 0^(0^(0...)) (n times), where we take 0^0 to be 1. - Natan Arie Consigli, May 02 2015
Euler transform and inverse Moebius transform of length 2 sequence [0, 1]. - Michael Somos, Feb 20 2024

A001227 Number of odd divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 4, 1, 2, 3, 2, 2, 4, 2, 2, 2, 3, 2, 4, 2, 2, 4, 2, 1, 4, 2, 4, 3, 2, 2, 4, 2, 2, 4, 2, 2, 6, 2, 2, 2, 3, 3, 4, 2, 2, 4, 4, 2, 4, 2, 2, 4, 2, 2, 6, 1, 4, 4, 2, 2, 4, 4, 2, 3, 2, 2, 6, 2, 4, 4, 2, 2, 5, 2, 2, 4, 4, 2, 4, 2, 2, 6, 4, 2, 4, 2, 4, 2, 2, 3, 6, 3, 2, 4, 2, 2, 8

Offset: 1

N. J. A. Sloane

Also (1) number of ways to write n as difference of two triangular numbers (A000217), see A136107; (2) number of ways to arrange n identical objects in a trapezoid. - Tom Verhoeff
Also number of partitions of n into consecutive positive integers including the trivial partition of length 1 (e.g., 9 = 2+3+4 or 4+5 or 9 so a(9)=3). (Useful for cribbage players.) See A069283. - Henry Bottomley, Apr 13 2000
This has been described as Sylvester's theorem, but to reduce ambiguity I suggest calling it Sylvester's enumeration. - Gus Wiseman, Oct 04 2022
a(n) is also the number of factors in the factorization of the Chebyshev polynomial of the first kind T_n(x). - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 28 2003
Number of factors in the factorization of the polynomial x^n+1 over the integers. See also A000005. - T. D. Noe, Apr 16 2003
a(n) = 1 if and only if n is a power of 2 (see A000079). - Lekraj Beedassy, Apr 12 2005
Number of occurrences of n in A049777. - Philippe Deléham, Jun 19 2005
For n odd, n is prime if and only if a(n) = 2. - George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Sep 10 2005
Also number of partitions of n such that if k is the largest part, then each of the parts 1,2,...,k-1 occurs exactly once. Example: a(9)=3 because we have [3,3,2,1],[2,2,2,2,1] and [1,1,1,1,1,1,1,1,1]. - Emeric Deutsch, Mar 07 2006
Also the number of factors of the n-th Lucas polynomial. - T. D. Noe, Mar 09 2006
Lengths of rows of triangle A182469;
Denoted by Delta_0(n) in Glaisher 1907. - Michael Somos, May 17 2013
Also the number of partitions p of n into distinct parts such that max(p) - min(p) < length(p). - Clark Kimberling, Apr 18 2014
Row sums of triangle A247795. - Reinhard Zumkeller, Sep 28 2014
Row sums of triangle A237048. - Omar E. Pol, Oct 24 2014
A069288(n) <= a(n). - Reinhard Zumkeller, Apr 05 2015
A000203, A000593 and this sequence have the same parity: A053866. - Omar E. Pol, May 14 2016
a(n) is equal to the number of ways to write 2*n-1 as (4*x + 2)*y + 4*x + 1 where x and y are nonnegative integers. Also a(n) is equal to the number of distinct values of k such that k/(2*n-1) + k divides (k/(2*n-1))^(k/(2*n-1)) + k, (k/(2*n-1))^k + k/(2*n-1) and k^(k/(2*n-1)) + k/(2*n-1). - Juri-Stepan Gerasimov, May 23 2016, Jul 15 2016
Also the number of odd divisors of n*2^m for m >= 0. - Juri-Stepan Gerasimov, Jul 15 2016
a(n) is odd if and only if n is a square or twice a square. - Juri-Stepan Gerasimov, Jul 17 2016
a(n) is also the number of subparts in the symmetric representation of sigma(n). For more information see A279387 and A237593. - Omar E. Pol, Nov 05 2016
a(n) is also the number of partitions of n into an odd number of equal parts. - Omar E. Pol, May 14 2017 [This follows from the g.f. Sum_{k >= 1} x^k/(1-x^(2*k)). - N. J. A. Sloane, Dec 03 2020]

			G.f. = q + q^2 + 2*q^3 + q^4 + 2*q^5 + 2*q^6 + 2*q^7 + q^8 + 3*q^9 + 2*q^10 + ...
From _Omar E. Pol_, Nov 30 2020: (Start)
For n = 9 there are three odd divisors of 9; they are [1, 3, 9]. On the other hand there are three partitions of 9 into consecutive parts: they are [9], [5, 4] and [4, 3, 2], so a(9) = 3.
Illustration of initial terms:
                              Diagram
   n   a(n)                         _
   1     1                        _|1|
   2     1                      _|1 _|
   3     2                    _|1  |1|
   4     1                  _|1   _| |
   5     2                _|1    |1 _|
   6     2              _|1     _| |1|
   7     2            _|1      |1  | |
   8     1          _|1       _|  _| |
   9     3        _|1        |1  |1 _|
  10     2      _|1         _|   | |1|
  11     2    _|1          |1   _| | |
  12     2   |1            |   |1  | |
...
a(n) is the number of horizontal line segments in the n-th level of the diagram. For more information see A286001. (End)

B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 487 Entry 47.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 306.
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4).
Ronald. L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics, 2nd ed. (Addison-Wesley, 1994), see exercise 2.30 on p. 65.
P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 28.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
K. S. Brown's Mathpages, Partitions into Consecutive Integers.
Atli Fannar Franklín, Pattern avoidance enumerated by inversions, arXiv:2410.07467 [math.CO], 2024. See pp. 2, 18.
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
A. Heiligenbrunner, Sum of adjacent numbers (in German).
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 11.
Gerzson Kéri, The factorization of compressed Chebyshev polynomials and other polynomials related to multiple-angle formulas, Annales Univ. Sci. Budapest (Hungary, 2022) Sect. Comp., Vol. 53, 93-108.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function tau_o(n).
M. A. Nyblom, On the representation of the integers as a difference of nonconsecutive triangular numbers, Fibonacci Quarterly 39:3 (2001), pp. 256-263.
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976.
N. J. A. Sloane, Transforms.
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2 (1999), Article 99.1.6.
Eric Weisstein's World of Mathematics, Binomial Number and Odd Divisor Function.
Eric Weisstein's World of Mathematics, q-Polygamma Function.
Index entries for "core" sequences.
Index entries for sequences mentioned by Glaisher.

Cf. A000005, A000079, A000593, A010054 (char. func.), A038547 (positions of first appearances), A050999, A051000, A051001, A051002, A051731, A054844, A069283, A069288, A109814, A115369, A118235, A118236, A125911, A136655, A183063, A183064, A237593, A247795, A272887, A273401, A279387, A286001.
Cf. A000203, A001620, A002162, A053866, A060831.
If this sequence counts gapless sets by sum (by Sylvester's enumeration), these sets are ranked by A073485 and A356956. See also A055932, A066311, A073491, A107428, A137921, A333217, A356224, A356841, A356845.
Dirichlet inverse is A327276.

a001227 = sum . a247795_row
-- Reinhard Zumkeller, Sep 28 2014, May 01 2012, Jul 25 2011

[NumberOfDivisors(n)/Valuation(2*n, 2): n in [1..100]]; // Vincenzo Librandi, Jun 02 2019

for n from 1 by 1 to 100 do s := 0: for d from 1 by 2 to n do if n mod d = 0 then s := s+1: fi: od: print(s); od:
A001227 := proc(n) local a,d;
    a := 1 ;
    for d in ifactors(n)[2] do
        if op(1,d) > 2 then
            a := a*(op(2,d)+1) ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Jun 18 2015

f[n_] := Block[{d = Divisors[n]}, Count[ OddQ[d], True]]; Table[ f[n], {n, 105}] (* Robert G. Wilson v, Aug 27 2004 *)
Table[Total[Mod[Divisors[n], 2]],{n,105}] (* Zak Seidov, Apr 16 2010 *)
f[n_] := Block[{d = DivisorSigma[0, n]}, If[ OddQ@ n, d, d - DivisorSigma[0, n/2]]]; Array[f, 105] (* Robert G. Wilson v *)
a[ n_] := Sum[  Mod[ d, 2], { d, Divisors[ n]}]; (* Michael Somos, May 17 2013 *)
a[ n_] := DivisorSum[ n, Mod[ #, 2] &]; (* Michael Somos, May 17 2013 *)
Count[Divisors[#],?OddQ]&/@Range[110] (* _Harvey P. Dale, Feb 15 2015 *)
(* using a262045 from A262045 to compute a(n) = number of subparts in the symmetric representation of sigma(n) *)
(* cl = current level, cs = current subparts count *)
a001227[n_] := Module[{cs=0, cl=0, i, wL, k}, wL=a262045[n]; k=Length[wL]; For[i=1, i<=k, i++, If[wL[[i]]>cl, cs++; cl++]; If[wL[[i]]Hartmut F. W. Hoft, Dec 16 2016 *)
a[n_] := DivisorSigma[0, n / 2^IntegerExponent[n, 2]]; Array[a, 100] (* Amiram Eldar, Jun 12 2022 *)

{a(n) = sumdiv(n, d, d%2)}; /* Michael Somos, Oct 06 2007 */

{a(n) = direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( 4, p) * X))[n]}; /* Michael Somos, Oct 06 2007 */

a(n)=numdiv(n>>valuation(n,2)) \\ Charles R Greathouse IV, Mar 16 2011

a(n)=sum(k=1,round(solve(x=1,n,x*(x+1)/2-n)),(k^2-k+2*n)%(2*k)==0) \\ Charles R Greathouse IV, May 31 2013

a(n)=sumdivmult(n,d,d%2) \\ Charles R Greathouse IV, Aug 29 2013

from functools import reduce
from operator import mul
from sympy import factorint
def A001227(n): return reduce(mul,(q+1 for p, q in factorint(n).items() if p > 2),1) # Chai Wah Wu, Mar 08 2021

def A001227(n): return len([1 for d in divisors(n) if is_odd(d)])
[A001227(n) for n in (1..80)]  # Peter Luschny, Feb 01 2012

Dirichlet g.f.: zeta(s)^2*(1-1/2^s).
Comment from N. J. A. Sloane, Dec 02 2020: (Start)
By counting the odd divisors f n in different ways, we get three different ways of writing the ordinary generating function.  It is:
A(x) =  x + x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + x^8 + 3*x^9 + 2*x^10 + ...
  = Sum_{k >= 1} x^(2*k-1)/(1-x^(2*k-1))
  = Sum_{k >= 1} x^k/(1-x^(2*k))
  = Sum_{k >= 1} x^(k*(k+1)/2)/(1-x^k)  [Ramanujan, 2nd notebook, p. 355.].
(This incorporates comments from Vladeta Jovovic, Oct 16 2002 and  Michael Somos, Oct 30 2005.) (End)
G.f.: x/(1-x) + Sum_{n>=1} x^(3*n)/(1-x^(2*n)), also L(x)-L(x^2) where L(x) = Sum_{n>=1} x^n/(1-x^n). - Joerg Arndt, Nov 06 2010
a(n) = A000005(n)/(A007814(n)+1) = A000005(n)/A001511(n).
Multiplicative with a(p^e) = 1 if p = 2; e+1 if p > 2. - David W. Wilson, Aug 01 2001
a(n) = A000005(A000265(n)). - Lekraj Beedassy, Jan 07 2005
Moebius transform is period 2 sequence [1, 0, ...] = A000035, which means a(n) is the Dirichlet convolution of A000035 and A057427.
a(n) = A113414(2*n). - N. J. A. Sloane, Jan 24 2006 (corrected Nov 10 2007)
a(n) = A001826(n) + A001842(n). - Reinhard Zumkeller, Apr 18 2006
Sequence = M*V = A115369 * A000005, where M = an infinite lower triangular matrix and V = A000005, d(n); as a vector: [1, 2, 2, 3, 2, 4, ...]. - Gary W. Adamson, Apr 15 2007
Equals A051731 * [1,0,1,0,1,...]; where A051731 is the inverse Mobius transform. - Gary W. Adamson, Nov 06 2007
a(n) = A000005(n) - A183063(n).
a(n) = d(n) if n is odd, or d(n) - d(n/2) if n is even, where d(n) is the number of divisors of n (A000005). (See the Weisstein page.) - Gary W. Adamson, Mar 15 2011
Dirichlet convolution of A000005 and A154955 (interpreted as a flat sequence). - R. J. Mathar, Jun 28 2011
a(A000079(n)) = 1; a(A057716(n)) > 1; a(A093641(n)) <= 2; a(A038550(n)) = 2; a(A105441(n)) > 2; a(A072502(n)) = 3. - Reinhard Zumkeller, May 01 2012
a(n) = 1 + A069283(n). - R. J. Mathar, Jun 18 2015
a(A002110(n)/2) = n, n >= 1. - Altug Alkan, Sep 29 2015
a(n*2^m) = a(n*2^i), a((2*j+1)^n) = n+1 for m >= 0, i >= 0 and j >= 0. a((2*x+1)^n) = a((2*y+1)^n) for positive x and y. - Juri-Stepan Gerasimov, Jul 17 2016
Conjectures: a(n) = A067742(n) + 2*A131576(n) = A082647(n) + A131576(n). - Omar E. Pol, Feb 15 2017
a(n) = A000005(2n) - A000005(n) = A099777(n)-A000005(n). - Danny Rorabaugh, Oct 03 2017
L.g.f.: -log(Product_{k>=1} (1 - x^(2*k-1))^(1/(2*k-1))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018
G.f.: (psi_{q^2}(1/2) + log(1-q^2))/log(q), where psi_q(z) is the q-digamma function. - Michael Somos, Jun 01 2019
a(n) = A003056(n) - A238005(n). - Omar E. Pol, Sep 12 2021
Sum_{k=1..n} a(k) ~ n*log(n)/2 + (gamma + log(2)/2 - 1/2)*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 27 2022
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A000005(k) = log(2) (A002162). - Amiram Eldar, Mar 01 2023
a(n) = Sum_{i=1..n} (-1)^(i+1)*A135539(n,i). - Ridouane Oudra, Apr 13 2023

A209229 Characteristic function of powers of 2, cf. A000079.

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Reinhard Zumkeller, Mar 06 2012

Essentially the same as A036987 (the Fredholm-Rueppel sequence).
Completely multiplicative with a(2^e) = 1, a(p^e) = 0 for odd primes p. - Mitch Harris, Apr 19 2005
Moebius transform of A001511. - R. J. Mathar, Jun 20 2014

			x + x^2 + x^4 + x^8 + x^16 + x^32 + x^64 + x^128 + x^256 + x^512 + x^1024 + ...

Michel Dekking, Michel Mendes France and Alf van der Poorten, "Folds", The Mathematical Intelligencer, Vol. 4, No. 3 (1982), pp. 130-138 & front cover, and Vol. 4, No. 4 (1982), pp. 173-181 (printed in two parts).
Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

Antti Karttunen, Table of n, a(n) for n = 0..65537 (terms 0..1000 from G. C. Greubel)
Index entries for characteristic functions
Index to divisibility sequences

Cf. A001511, A029837 (partial sums), A087003 (moebius transform), A090678, A104977, A154955 (Dirichlet inverse).

Cf. A069517, A154269, A201219, A255738.

int a (unsigned long n) { return n & !(n & (n-1)); } /* Charles R Greathouse IV, Sep 15 2012 */

a209229 n | n < 2 = n
          | n > 1 = if m > 0 then 0 else a209229 n'
          where (n',m) = divMod n 2

A209229 := proc(n)
    if n <= 0 then
        0 ;
    elif n = 1 then
        1;
    elif type (n,'odd') or A001221(n) > 1 then
        0 ;
    else
        1;
    end if;
end proc:
seq(A209229(n),n=0..40) ; # R. J. Mathar, Jan 07 2021

a[n_] := Boole[n == 2^IntegerExponent[n, 2]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 06 2014 *)
Table[If[IntegerQ[Log[2,n]],1,0],{n,0,100}] (* Harvey P. Dale, Jun 24 2018 *)

a(n)=n==1<Charles R Greathouse IV, Mar 07 2012

{a(n) = if( n<2 || n%2, n==1, isprimepower(n) > 0)} \\ Michael Somos, Jan 03 2013

def A209229(n): return int(not(n&-n)^n) if n else 0 # Chai Wah Wu, Jul 08 2022

a(A000079(n)) = 1; a(A057716(n)) = 0.
a(n+1) = A036987(n).
a(n) = if n < 2 then n else (if n is even then a(n/2) else 0).
The generating function g(x) satisfies g(x) - g(x^2) = x. - Joerg Arndt, May 11 2010
Dirichlet g.f.: 1/(1 - 2^(-s)). - R. J. Mathar, Mar 07 2012
G.f.: x / (1 - x / (1 + x / (1 + x / (1 - x / (1 + x / (1 - x / ...)))))) = x / (1 + b(1) * x / (1 + b(2) * x / (1 + b(3) * x / ...))) where b(n) = (-1)^ A090678(n+1). - Michael Somos, Jan 03 2013
With a(0) = 0 removed is convolution inverse of A104977. - Michael Somos, Jan 03 2013
From Antti Karttunen, Nov 19 2017: (Start)
a(n) = abs(A154269(n)).
For n > 1, a(n) = A069517(n)/2 = 2 - A201219(n). (End)
a(n) = A048298(n)/n. - R. J. Mathar, Jan 07 2021
a(n) = floor((2^n)/n) - floor((2^n - 1)/n), for n>=1. - Ridouane Oudra, Oct 15 2021

A108299 Triangle read by rows, 0 <= k <= n: T(n,k) = binomial(n-[(k+1)/2],[k/2])*(-1)^[(k+1)/2].

Original entry on oeis.org

1, 1, -1, 1, -1, -1, 1, -1, -2, 1, 1, -1, -3, 2, 1, 1, -1, -4, 3, 3, -1, 1, -1, -5, 4, 6, -3, -1, 1, -1, -6, 5, 10, -6, -4, 1, 1, -1, -7, 6, 15, -10, -10, 4, 1, 1, -1, -8, 7, 21, -15, -20, 10, 5, -1, 1, -1, -9, 8, 28, -21, -35, 20, 15, -5, -1, 1, -1, -10, 9, 36, -28, -56, 35, 35, -15, -6, 1, 1, -1, -11, 10, 45, -36, -84, 56, 70

Offset: 0

Reinhard Zumkeller, Jun 01 2005

Matrix inverse of A124645.
Let L(n,x) = Sum_{k=0..n} T(n,k)*x^(n-k) and Pi=3.14...:
L(n,x) = Product_{k=1..n} (x - 2*cos((2*k-1)*Pi/(2*n+1)));
Sum_{k=0..n} T(n,k) = L(n,1) = A010892(n+1);
Sum_{k=0..n} abs(T(n,k)) = A000045(n+2);
abs(T(n,k)) = A065941(n,k), T(n,k) = A065941(n,k)*A087960(k);
T(2*n,k) + T(2*n+1,k+1) = 0 for 0 <= k <= 2*n;
T(n,0) = A000012(n) = 1; T(n,1) = -1 for n > 0;
T(n,2) = -(n-1) for n > 1; T(n,3) = A000027(n)=n for n > 2;
T(n,4) = A000217(n-3) for n > 3; T(n,5) = -A000217(n-4) for n > 4;
T(n,6) = -A000292(n-5) for n > 5; T(n,7) = A000292(n-6) for n > 6;
T(n,n-3) = A058187(n-3)*(-1)^floor(n/2) for n > 2;
T(n,n-2) = A008805(n-2)*(-1)^floor((n+1)/2) for n > 1;
T(n,n-1) = A008619(n-1)*(-1)^floor(n/2) for n > 0;
T(n,n) = L(n,0) = (-1)^floor((n+1)/2);
L(n,1) = A010892(n+1); L(n,-1) = A061347(n+2);
L(n,2) = 1; L(n,-2) = A005408(n)*(-1)^n;
L(n,3) = A001519(n); L(n,-3) = A002878(n)*(-1)^n;
L(n,4) = A001835(n+1); L(n,-4) = A001834(n)*(-1)^n;
L(n,5) = A004253(n); L(n,-5) = A030221(n)*(-1)^n;
L(n,6) = A001653(n); L(n,-6) = A002315(n)*(-1)^n;
L(n,7) = A049685(n); L(n,-7) = A033890(n)*(-1)^n;
L(n,8) = A070997(n); L(n,-8) = A057080(n)*(-1)^n;
L(n,9) = A070998(n); L(n,-9) = A057081(n)*(-1)^n;
L(n,10) = A072256(n+1); L(n,-10) = A054320(n)*(-1)^n;
L(n,11) = A078922(n+1); L(n,-11) = A097783(n)*(-1)^n;
L(n,12) = A077417(n); L(n,-12) = A077416(n)*(-1)^n;
L(n,13) = A085260(n);
L(n,14) = A001570(n); L(n,-14) = A028230(n)*(-1)^n;
L(n,n) = A108366(n); L(n,-n) = A108367(n).
Row n of the matrix inverse (A124645) has g.f.: x^floor(n/2)*(1-x)^(n-floor(n/2)). - Paul D. Hanna, Jun 12 2005
From L. Edson Jeffery, Mar 12 2011: (Start)
Conjecture: Let N=2*n+1, with n > 2. Then T(n,k) (0 <= k <= n) gives the k-th coefficient in the characteristic function p_N(x)=0, of degree n in x, for the n X n tridiagonal unit-primitive matrix G_N (see [Jeffery]) of the form
  G_N=A_{N,1}=
  (0 1 0 ... 0)
  (1 0 1 0 ... 0)
  (0 1 0 1 0 ... 0)
  ...
  (0 ... 0 1 0 1)
  (0 ... 0 1 1),
with solutions phi_j = 2*cos((2*j-1)*Pi/N), j=1,2,...,n. For example, for n=3,
  G_7=A_{7,1}=
  (0 1 0)
  (1 0 1)
  (0 1 1).
We have {T(3,k)}=(1,-1,-2,1), while the characteristic function of G_7 is p(x) = x^3-x^2-2*x+1 = 0, with solutions phi_j = 2*cos((2*j-1)*Pi/7), j=1,2,3. (End)
The triangle sums, see A180662 for their definitions, link A108299 with several sequences, see the crossrefs. - Johannes W. Meijer, Aug 08 2011
The roots to the polynomials are chaotic using iterates of the operation (x^2 - 2), with cycle lengths L and initial seeds returning to the same term or (-1)* the seed. Periodic cycle lengths L are shown in A003558 such that for the polynomial represented by row r, the cycle length L is A003558(r-1). The matrices corresponding to the rows as characteristic polynomials are likewise chaotic [cf. Kappraff et al., 2005] with the same cycle lengths but substituting 2*I for the "2" in (x^2 - 2), where I = the Identity matrix. For example, the roots to x^3 - x^2 - 2x + 1 = 0 are 1.801937..., -1.246979..., and 0.445041... With 1.801937... as the initial seed and using (x^2 - 2), we obtain the 3-period trajectory of 8.801937... -> 1.246979... -> -0.445041... (returning to -1.801937...). We note that A003558(2) = 3. The corresponding matrix M is: [0,1,0; 1,0,1; 0,1,1,]. Using seed M with (x^2 - 2*I), we obtain the 3-period with the cycle completed at (-1)*M. - Gary W. Adamson, Feb 07 2012

			Triangle begins:
  1;
  1,  -1;
  1,  -1,  -1;
  1,  -1,  -2,   1;
  1,  -1,  -3,   2,   1;
  1,  -1,  -4,   3,   3,  -1;
  1,  -1,  -5,   4,   6,  -3,  -1;
  1,  -1,  -6,   5,  10,  -6,  -4,   1;
  1,  -1,  -7,   6,  15, -10, -10,   4,   1;
  1,  -1,  -8,   7,  21, -15, -20,  10,   5,  -1;
  1,  -1,  -9,   8,  28, -21, -35,  20,  15,  -5,  -1;
  1,  -1, -10,   9,  36, -28, -56,  35,  35, -15,  -6,   1;
  ...

Friedrich L. Bauer, 'De Moivre und Lagrange: Cosinus eines rationalen Vielfachen von Pi', Informatik Spektrum 28 (Springer, 2005).
Jay Kappraff, S. Jablan, G. Adamson, & R. Sazdonovich: "Golden Fields, Generalized Fibonacci Sequences, & Chaotic Matrices"; FORMA, Vol 19, No 4, (2005).

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
Henry W. Gould, A Variant of Pascal's Triangle, Corrections, The Fibonacci Quarterly, Vol. 3, Nr. 4, Dec. 1965, p. 257-271.
L. Edson Jeffery, Unit-primitive matrices.
Ju, Hyeong-Kwan On the sequence generated by a certain type of matrices. Honam Math. J. 39, No. 4, 665-675 (2017).
Michelle Rudolph-Lilith, On the Product Representation of Number Sequences, with Application to the Fibonacci Family, arXiv preprint arXiv:1508.07894 [math.NT], 2015.
Frank Ruskey and Carla Savage, Gray codes for set partitions and restricted growth tails, Australasian Journal of Combinatorics, Volume 10(1994), pp. 85-96. See Table 1 p. 95.

Cf. A049310, A039961, A124645 (matrix inverse).
Triangle sums (see the comments): A193884 (Kn11), A154955 (Kn21), A087960 (Kn22), A000007 (Kn3), A010892 (Fi1), A134668 (Fi2), A078031 (Ca2), A193669 (Gi1), A001519 (Gi3), A193885 (Ze1), A050935 (Ze3). - Johannes W. Meijer, Aug 08 2011
Cf. A003558.
Cf. A033999, A059841.

a108299 n k = a108299_tabl !! n !! k
a108299_row n = a108299_tabl !! n
a108299_tabl = [1] : iterate (\row ->
   zipWith (+) (zipWith (*) ([0] ++ row) a033999_list)
               (zipWith (*) (row ++ [0]) a059841_list)) [1,-1]
-- Reinhard Zumkeller, May 06 2012

A108299 := proc(n,k): binomial(n-floor((k+1)/2), floor(k/2))*(-1)^floor((k+1)/2) end: seq(seq(A108299 (n,k), k=0..n), n=0..11); # Johannes W. Meijer, Aug 08 2011

t[n_, k_?EvenQ] := I^k*Binomial[n-k/2, k/2]; t[n_, k_?OddQ] := -I^(k-1)*Binomial[n+(1-k)/2-1, (k-1)/2]; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 16 2013 *)

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

T(n,k) = binomial(n-floor((k+1)/2),floor(k/2))*(-1)^floor((k+1)/2).
T(n+1, k) = if sign(T(n, k-1))=sign(T(n, k)) then T(n, k-1)+T(n, k) else -T(n, k-1) for 0 < k < n, T(n, 0) = 1, T(n, n) = (-1)^floor((n+1)/2).
G.f.: A(x, y) = (1 - x*y)/(1 - x + x^2*y^2). - Paul D. Hanna, Jun 12 2005
The generating polynomial (in z) of row n >= 0 is (u^(2*n+1) + v^(2*n+1))/(u + v), where u and v are defined by u^2 + v^2 = 1 and u*v = z. - Emeric Deutsch, Jun 16 2011
From Johannes W. Meijer, Aug 08 2011: (Start)
abs(T(n,k)) = A065941(n,k) = abs(A187660(n,n-k));
T(n,n-k) = A130777(n,k); abs(T(n,n-k)) = A046854(n,k) = abs(A066170(n,k)). (End)

Corrected and edited by Philippe Deléham, Oct 20 2008

A007331 Fourier coefficients of E_{infinity,4}.

Original entry on oeis.org

0, 1, 8, 28, 64, 126, 224, 344, 512, 757, 1008, 1332, 1792, 2198, 2752, 3528, 4096, 4914, 6056, 6860, 8064, 9632, 10656, 12168, 14336, 15751, 17584, 20440, 22016, 24390, 28224, 29792, 32768, 37296, 39312, 43344, 48448, 50654, 54880, 61544, 64512

Offset: 0

N. J. A. Sloane, Mira Bernstein

E_{infinity,4} is the unique normalized weight-4 modular form for Gamma_0(2) with simple zeros at i*infinity. Since this has level 2, it is not a cusp form, in contrast to A002408.
a(n+1) is the number of representations of n as a sum of 8 triangular numbers (from A000217). See the Ono et al. link, Theorem 5.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(n) gives the sum of cubes of divisors d of n such that n/d is odd. This is called sigma^#3(n) in the Ono et al. link. See a formula below. - _Wolfdieter Lang, Jan 12 2017

			G.f. = q + 8*q^2 + 28*q^3 + 64*q^4 + 126*q^5 + 224*q^6 + 344*q^7 + 512*q^8 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 139, Ex (ii).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1001 from T. D. Noe)
B. Brent, Quadratic Minima and Modular Forms, Experimental Mathematics, v.7 no.3, 257-274.
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 187.
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94. Theorem 5.
H. Rosengren, Sums of triangular numbers from the Frobenius determinant, arXiv:math/0504272 [math.NT], 2005.
Index entries for sequences mentioned by Glaisher.

Cf. A002408, A004017, A035016, A045825, A076577, A096960, A222072.

Number of ways of writing n as a sum of k triangular numbers, for k=1,...: A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787, A014809, A076577.

Basis( ModularForms( Gamma0(2), 4), 10) [2]; /* Michael Somos, May 27 2014 */

nmax:=40: seq(coeff(series(x*(product((1-x^k)^8*(1+x^k)^16, k=1..nmax)), x, n+1), x, n), n=0..nmax); # Vaclav Kotesovec, Oct 14 2015

Prepend[Table[Plus @@ (Select[Divisors[k + 1], OddQ[(k + 1)/#] &]^3), {k, 0, 39}], 0] (* Ant King, Dec 04 2010 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)]^8 / 256, {q, 0, n}]; (* Michael Somos, Jun 04 2013 *)
a[ n_] := If[ n < 1, 0, Sum[ d^3 Boole[ OddQ[ n/d]], {d, Divisors[ n]}]]; (* Michael Somos, Jun 04 2013 *)
f[n_] := Total[(2n/Select[ Divisors[ 2n], Mod[#, 4] == 2 &])^3]; Flatten[{0, Array[f, 40] }] (* Robert G. Wilson v, Mar 26 2015 *)
nmax=60; CoefficientList[Series[x*Product[(1-x^k)^8 * (1+x^k)^16, {k,1,nmax}],{x,0,nmax}], x] (* Vaclav Kotesovec, Oct 14 2015 *)
QP = QPochhammer; s = q * (QP[-1, q]/2)^16 * QP[q]^8 + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)

{a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^3))}; /* Michael Somos, May 31 2005 */

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / eta(x + A))^8, n))}; /* Michael Somos, May 31 2005 */

a(n)=my(e=valuation(n,2)); 8^e * sigma(n/2^e, 3) \\ Charles R Greathouse IV, Sep 09 2014

from sympy import divisors
def a(n):
    return 0 if n == 0 else sum(((n//d)%2)*d**3 for d in divisors(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 24 2017

ModularForms( Gamma0(2), 4, prec=33).1; # Michael Somos, Jun 04 2013

G.f.: q * Product_{k>=1} (1-q^k)^8 * (1+q^k)^16. - corrected by Vaclav Kotesovec, Oct 14 2015

a(n) = Sum_{0

G.f.: Sum_{n>0} n^3*x^n/(1-x^(2*n)). - Vladeta Jovovic, Oct 24 2002

Expansion of Jacobi theta constant theta_2(q)^8 / 256 in powers of q.

Expansion of eta(q^2)^16 / eta(q)^8 in powers of q. - Michael Somos, May 31 2005

Expansion of x * psi(x)^8 in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Jan 15 2012

Expansion of (Q(x) - Q(x^2)) / 240 in powers of x where Q() is a Ramanujan Lambert series. - Michael Somos, Jan 15 2012

Expansion of E_{gamma,2}^2 * E_{0,4} in powers of q.

Euler transform of period 2 sequence [8, -8, ...]. - Michael Somos, May 31 2005

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 - u^2*w + 16*u*v*w - 32*v^2*w + 256*v*w^2. - Michael Somos, May 31 2005

G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 16^(-1) (t / i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A035016. - Michael Somos, Jan 11 2009

Multiplicative with a(2^e) = 2^(3e), a(p^e) = (p^(3(e+1))-1)/(p^3-1). - Mitch Harris, Jun 13 2005

Dirichlet convolution of A154955 by A001158. Dirichlet g.f. zeta(s)*zeta(s-3)*(1-1/2^s). - R. J. Mathar, Mar 31 2011

A002408(n) = -(-1)^n * a(n).

Convolution square of A008438. - Michael Somos, Jun 15 2014

a(1) = 1, a(n) = (8/(n-1))*Sum_{k=1..n-1} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017

Sum_{k=1..n} a(k) ~ c * n^4, where c = Pi^4/384 = 0.253669... (A222072). - Amiram Eldar, Oct 19 2022

Additional comments from Barry Brent (barryb(AT)primenet.com)
Wrong Maple program replaced by Vaclav Kotesovec, Oct 14 2015
a(0)=0 prepended by Vaclav Kotesovec, Oct 14 2015

A181983 a(n) = (-1)^(n+1) * n.

Original entry on oeis.org

0, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15, -16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -26, 27, -28, 29, -30, 31, -32, 33, -34, 35, -36, 37, -38, 39, -40, 41, -42, 43, -44, 45, -46, 47, -48, 49, -50, 51, -52, 53, -54, 55, -56, 57, -58, 59

Offset: 0

Michael Somos, Apr 04 2012

This is the Lucas U(-2,1) sequence. - R. J. Mathar, Jan 08 2013
Apparently the Mobius transform of A002129. - R. J. Mathar, Jan 08 2013
For n>0, a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = max(i,j) for 1 <= i,j <= n. - Enrique Pérez Herrero, Jan 14 2013
The sums of the terms of this sequence is the divergent series 1 - 2 + 3 - 4 + ... . Euler summed it to 1/4 which was one of the first examples of summing divergent series. - Michael Somos, Jun 05 2013

			G.f. = x - 2*x^2 + 3*x^3 - 4*x^4 + 5*x^5 - 6*x^6 + 7*x^7 - 8*x^8 + 9*x^9 + ...

Paulo Ribenboim, My Numbers, My Friends: Popular Lectures on Number Theory, Springer-Verlag, NY, 2000, p. 6.

Enrique Pérez Herrero, Max Determinant, 2013.
Wikipedia, 1 - 2 + 3 - 4 + ....
Wikipedia, Lucas sequence.
Index entries for linear recurrences with constant coefficients, signature (-2,-1).
Index entries for Lucas sequences.

Cf. A000108, A000169, A001057, A001477, A001787, A002129, A038608, A049347, A154955, A274922.

a181983 = negate . a038608
a181983_list = [0, 1] ++ map negate
   (zipWith (+) a181983_list (map (* 2) $ tail a181983_list))
-- Reinhard Zumkeller, Mar 20 2013

[(-1)^(n+1)*n: n in [0..30]]; // G. C. Greubel, Aug 11 2018

A181983:=n->-(-1)^n * n; seq(A181983(n), n=0..100); # Wesley Ivan Hurt, Feb 26 2014

a[ n_] := -(-1)^n n;
a[ n_] := Sign[n] SeriesCoefficient[ x / (1 + x)^2, {x, 0, Abs @n}];
a[ n_] := Sign[n] (Abs @n)! SeriesCoefficient[ x / Exp[ x], {x, 0, Abs @n}];
CoefficientList[Series[x/(1+x)^2,{x,0,60}],x] (* or *) LinearRecurrence[{-2,-1},{0,1},60] (* or *) Table[If[OddQ[n],n,-n],{n,0,60}]  (* Harvey P. Dale, Apr 22 2022 *)

PARI
```
{a(n) = -(-1)^n * n};
```

G.f.: x / (1 + x)^2.
E.g.f.: x / exp(x).
a(n) = -a(-n) = -(-1)^n * A001477(n) for all n in Z.
a(n+1) = p(n+1) where p(x) is the unique degree-n polynomial such that p(k) = Bernoulli(k) for k = 0, 1, ..., n.
A001787(n) = p(0) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 1, ..., n+1. - Michael Somos, Jun 05 2013
Euler transform of length 2 sequence [-2, 2].
Series reversion of g.f. is A000108(n) (Catalan numbers) with a(0)=0.
Series reversion of e.g.f. is A000169. INVERT transform omitting a(0)=0 is A049347. PSUM transform is A001057. BINOMIAL transform is A154955. - Michael Somos, Jun 05 2013
n * a(n) = A162395(n). - Michael Somos, Jun 05 2013
a(n) = - A038608(n). - Reinhard Zumkeller, Mar 20 2013
a(n+2) = a(n) - 2*(-1)^n. - G. C. Greubel, Aug 11 2018
a(n) = - A274922(n) if n>0. - Michael Somos, Sep 24 2019
From Amiram Eldar, Oct 24 2023: (Start)
Multiplicative with a(2^e) = -2^e, and a(p^e) = p^e for an odd prime p.
Dirichlet g.f.: zeta(s-1) * (1-2^(2-s)). (End)

A132013 T(n,j) for an iterated mixed order Laguerre transform. Coefficients of the normalized generalized Laguerre polynomials (-1)^nn!L(n,1-n,x).

Original entry on oeis.org

1, -1, 1, 0, -2, 1, 0, 0, -3, 1, 0, 0, 0, -4, 1, 0, 0, 0, 0, -5, 1, 0, 0, 0, 0, 0, -6, 1, 0, 0, 0, 0, 0, 0, -7, 1, 0, 0, 0, 0, 0, 0, 0, -8, 1, 0, 0, 0, 0, 0, 0, 0, 0, -9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -11, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -12, 1

Offset: 0

Tom Copeland, Oct 30 2007

The matrix operation b = T*a can be characterized in several ways in terms of the coefficients a(n) and b(n), their o.g.f.s A(x) and B(x), or e.g.f.s EA(x) and EB(x).
1) b(0) = a(0), b(n) = a(n) - n*a(n-1) for n > 0
2) b(n) = n! Lag{n,(.)!*Lag[.,a(.),0],-1}, umbrally
3) b(n) = n! Sum_{j=0..min(1,n)} (-1)^j * binomial(n,j)*a(n-j)/(n-j)!
4) b(n) = (-1)^n n! Lag(n,a(.),1-n)
5) B(x) = (1-xDx) A(x) = [1-x*Lag(1,-xD:,0)] A(x)
6) EB(x) = (1-x) EA(x),
where D is the derivative w.r.t. x and Lag(n,x,m) is the associated Laguerre polynomial of order m. These formulas are easily generalized for repeated applications of the operator.
c = (1,-1,0,0,0,...) is the sequence associated to T under the list partition transform and the associated operations described in A133314. The reciprocal sequence is d = (0!,1!,2!,3!,4!,...).
Consequently, the inverse of T is TI(n,k) = binomial(n,k)*d(n-k) = A094587, which has the property that the terms at and below TI(m,m) are the associated sequence under the list partition transform for the inverse for T^(m+1) for m=0,1,2,3,... .
Row sums of T = [formula 3 with all a(n) = 1] = [binomial transform of c] = [coefficients of B(x) with A(x) = 1/(1-x)] = A024000 = (1,0,-1,-2,-3,...), with e.g.f. = [EB(x) with EA(x) = exp(x)] = (1-x) * exp(x) = exp(x)*exp(c(.)*x) = exp[(1+c(.))*x].
Alternating row sums of T = [formula 3 with all a(n) = (-1)^n] = [finite differences of c] = [coefficients of B(x) with A(x) = 1/(1+x)] = (1,-2,3,-4,...), with e.g.f. = [EB(x) with EA(x) = exp(-x)] = (1-x) * exp(-x) = exp(-x)*exp(c(.)*x) = exp[-(1-c(.))*x].
An e.g.f. for the o.g.f.s for repeated applications of T on A(x) is given by
exp[t*(1-xDx)] A(x) = e^t * Sum_{n=0,1,...} (-t*x)^n * Lag(n,-:xD:,0) A(x)
= e^t * exp{[-t*u/(1+t*u)]*:xD:} / (1+t*u) A(x) (eval. at u=x)
= e^t * A[x/(1+t*x)]/(1+t*x) .
See A132014 for more notes on repeated applications.

			First few rows of the triangle are
   1;
  -1,  1;
   0, -2,  1;
   0,  0, -3,  1;
   0,  0,  0, -4,  1;
   0,  0,  0,  0, -5,  1;
   0,  0,  0,  0,  0, -6,  1;
   0,  0,  0,  0,  0,  0, -7,  1;

G. C. Greubel, Rows n=0..100 of triangle, flattened
T. Copeland, Compositional inverse operators and Sheffer sequences, 2016.
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) #09.8.3.
Wikipedia, Appell sequence

Cf. A235706, A132382, A132159, A094587, A132014, A154955, A235706.

c := n -> `if`(n=0,1,`if`(n=1,-1,0)):
T := (n,k) -> binomial(n,k)*c(n-k); # Peter Luschny, Nov 14 2016

Table[PadLeft[{-n, 1}, n+1], {n, 0, 13}] // Flatten (* Jean-François Alcover, Apr 29 2014 *)

row(n) = Vecrev((-1)^n*n!*pollaguerre(n, 1-n)); \\ Michel Marcus, Jul 26 2021

T(n,k) = binomial(n,k)*c(n-k), with the sequence c defined in the comments.
E.g.f.: exp(x*y)(1-x), which implies the row polynomials form an Appell sequence. More relations can be found in A132382. - Tom Copeland, Dec 03 2013
From Tom Copeland, Apr 21 2014: (Start)
Change notation letting L(n,m,x) = Lag(n,x,m).
Row polynomials: (-1)^n*n!*L(n,1-n,x) = -x^(n-1)*L(1,n-1,x) =
(-1)^n*(1/(1-n)!)*K(-n,1-n+1,x) where K is Kummer's confluent hypergeometric function (as a limit of n+s as s tends to zero).
For the row polynomials, the lowering operator = d/dx and the raising operator = x - 1/(1-D).
T = I - A132440 = 2*I - exp[A238385-I] = signed exp[A238385-I], where I = identity matrix.
Operationally, (-1)^n*n!*L(n,1-n,-:xD:) = -x^(n-1)*:Dx:^n*x^(1-n) = (-1)^n*x^(-1)*:xD:^n*x = (-1)^n*n!*binomial(xD+1,n) = (-1)^n*n!*binomial(1,n)*K(-n,1-n+1,-:xD:) where :AB:^n = A^n*B^n for any two operators. Cf. A235706. (End)
The unsigned row polynomials have e.g.f. (1+t)e^(xt) = exp(t*p.(x)), umbrally, and p_n(x) = (1+D) x^n. With q_n(x) the row polynomials of A094587, p_n(x) = u_n(q.(v.(x))), umbrally, where u_n(x) = (-1)^n v_n(-x) = (-1)^n Lah_n(x), the Lah polynomials with e.g.f. exp[x*t/(t-1)]. This has the matrix form unsigned [T] = [p] = [u]*[q]*[v]. Conversely, q_n(x) = v_n (p.(u.(x))). - Tom Copeland, Nov 10 2016
n-th row polynomial: n!*Sum_{k = 0..n} (-1)^k*binomial(n,k)*Lag(k,1,x). - Peter Bala, Jul 25 2021

Title modified by Tom Copeland, Apr 21 2014

A201780 Riordan array ((1-x)^2/(1-2x), x/(1-2x)).

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 2, 5, 4, 1, 4, 12, 13, 6, 1, 8, 28, 38, 25, 8, 1, 16, 64, 104, 88, 41, 10, 1, 32, 144, 272, 280, 170, 61, 12, 1, 64, 320, 688, 832, 620, 292, 85, 14, 1, 128, 704, 1696, 2352, 2072, 1204, 462, 113, 16, 1

Offset: 0

Philippe Deléham, Dec 05 2011

Diagonals ascending: 1, 0, 1, 1, 2, 2, 4, 5, 1, 8, 12, 4, ... (see A201509).

			Triangle begins:
  1;
  0,  1;
  1,  2,  1;
  2,  5,  4,  1;
  4, 12, 13,  6,  1;
  8, 28, 38, 25,  8,  1;

Benjamin Braun, W. K. Hough, Matching and Independence Complexes Related to Small Grids, arXiv preprint arXiv:1606.01204 [math.CO], 2016.
Wesley K. Hough, On Independence, Matching, and Homomorphism Complexes, (2017), Theses and Dissertations--Mathematics, 42.

Diagonals: A000012, A005843, A001844, A035597,
Columns: A034008, A045623, A084851, A055585,
Row sums: A052156

CoefficientList[#, y]& /@ CoefficientList[(1-x)^2/(1-(y+2)*x) + O[x]^10, x] // Flatten (* Jean-François Alcover, Nov 03 2018 *)

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) with T(0,0) = 0, T(1,0) = 0, T(2,0) = 0 and T(n,k)= 0 if k < 0 or if n < k.
Sum_{k=0..n} T(n,k)*x^k = A154955(n+1), A034008(n), A052156(n), A055841(n), A055842(n), A055846(n), A055270(n), A055847(n), A055995(n), A055996(n), A056002(n), A056116(n) for x = -1,0,1,2,3,4,5,6,7,8,9,10 respectively.
G.f.: (1-x)^2/(1-(y+2)*x).

A100218 Riordan array ((1-2*x)/(1-x), (1-x)).

Original entry on oeis.org

1, -1, 1, -1, -2, 1, -1, 0, -3, 1, -1, 0, 2, -4, 1, -1, 0, 0, 5, -5, 1, -1, 0, 0, -2, 9, -6, 1, -1, 0, 0, 0, -7, 14, -7, 1, -1, 0, 0, 0, 2, -16, 20, -8, 1, -1, 0, 0, 0, 0, 9, -30, 27, -9, 1, -1, 0, 0, 0, 0, -2, 25, -50, 35, -10, 1, -1, 0, 0, 0, 0, 0, -11, 55, -77, 44, -11, 1, -1, 0, 0, 0, 0, 0, 2, -36, 105, -112, 54, -12, 1

Offset: 0

Paul Barry, Nov 08 2004

			Triangle begins as:
   1;
  -1,  1;
  -1, -2,  1;
  -1,  0, -3,  1;
  -1,  0,  2, -4,  1;
  -1,  0,  0,  5, -5,   1;
  -1,  0,  0, -2,  9,  -6,   1;
  -1,  0,  0,  0, -7,  14,  -7,  1;
  -1,  0,  0,  0,  2, -16,  20, -8,  1;
  -1,  0,  0,  0,  0,   9, -30, 27, -9,  1;

G. C. Greubel, Rows n = 0..49 of the triangle, flattened, flattened
P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.

Cf. A000007, A000027, A077961, A080956, A098601.
Cf. A154955, A157142, A280560, A355021.
Row sums are A100219.
Matrix inverse of A100100.
Apart from signs, same as A098599.
Very similar to triangle A111125.

A100218:= func< n,k | n eq 0 select 1 else (-1)^(n+k)*(Binomial(k,n-k) + Binomial(k-1,n-k-1)) >;
[A100218(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Mar 28 2024

T[0,0]:= 1; T[1,1]:= 1; T[1,0]:= -1; T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, T[n- 1,k] +T[n-1,k-1] -2*T[n-2,k-1] +T[n-3,k-1]]; Table[T[n,k], {n,0,14}, {k,0,n} ]//Flatten (* G. C. Greubel, Mar 13 2017 *)

def A100218(n,k): return 1 if n==0 else (-1)^(n+k)*(binomial(k,n-k) + binomial(k-1,n-k-1))
flatten([[A100218(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Mar 28 2024

Sum_{k=0..n} T(n, k) = A100219(n) (row sums).
Number triangle T(n, k) = (-1)^(n-k)*(binomial(k, n-k) + binomial(k-1, n-k-1)), with T(0, 0) = 1. - Paul Barry, Nov 09 2004
T(n,k) = T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k-1) + T(n-3,k-1), T(0,0)=1, T(1,0)=-1, T(1,1)=1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Jan 09 2014
From G. C. Greubel, Mar 28 2024: (Start)
T(n, n-1) = A000027(n), n >= 1.
T(n, n-2) = -A080956(n-1), n >= 2.
T(2*n, n) = A280560(n).
T(2*n-1, n) = A157142(n-1), n >= 1.
T(2*n+1, n) = -A000007(n) = A154955(n+2).
T(3*n, n) = T(4*n, n) = A000007(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A355021(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n*A098601(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = -1 + 2*A077961(n) + A077961(n-2). (End)
From Peter Bala, Apr 28 2024: (Start)
This Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = x*(1 - x) and hence belongs to the hitting time subgroup of the Riordan group (see Peart and Woan for properties of this subgroup).
T(n,k) = [x^(n-k)] (1/c(x))^n, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. In general the (n, k)-th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)

A132339 Array T(n, k) = (-1)^(n+k)(n+k-2)!(2n+2k-2)!/(n!k!(2n-1)!(2*k-1)!), with T(0, 0) = 1, T(0, 1) = T(1, 0) = -1, read by antidiagonals.

Original entry on oeis.org

1, -1, -1, 0, 2, 0, 0, -2, -2, 0, 0, 2, 10, 2, 0, 0, -2, -28, -28, -2, 0, 0, 2, 60, 168, 60, 2, 0, 0, -2, -110, -660, -660, -110, -2, 0, 0, 2, 182, 2002, 4290, 2002, 182, 2, 0, 0, -2, -280, -5096, -20020, -20020, -5096, -280, -2, 0, 0, 2, 408, 11424, 74256, 136136, 74256, 11424, 408, 2, 0

Offset: 0

N. J. A. Sloane, Nov 08 2007

			Array (T(n,k)) begins:
   1, -1,    0,     0,       0,       0,         0 ... A154955(k)
  -1,  2,   -2,     2,      -2,       2,        -2 ... (-1)^(k+1)*A040000(k)
   0, -2,   10,   -28,      60,    -110,       182 ... (-1)^k*A006331(k)
   0,  2,  -28,   168,    -660,    2002,     -5096 ... (-1)^k*A006332(k)
   0, -2,   60,  -660,    4290,  -20020,     74256 ... (-1)^k*A006333(k)
   0,  2, -110,  2002,  -20020,  136136,   -705432 ... (-1)^k*A006334(k)
   0, -2,  182, -5096,   74256, -705432,   4938024 ...
   0,  2, -280, 11424, -232560, 2984520, -27457584 ...
Antidiagonal (A(n,k)) triangle begins as:
   1;
  -1, -1;
   0,  2,    0;
   0, -2,   -2,     0;
   0,  2,   10,     2,      0;
   0, -2,  -28,   -28,     -2,      0;
   0,  2,   60,   168,     60,      2,     0;
   0, -2, -110,  -660,   -660,   -110,    -2,     0;
   0,  2,  182,  2002,   4290,   2002,   182,     2,   0;
   0, -2, -280, -5096, -20020, -20020, -5096,  -280,  -2,   0;
   0,  2,  408, 11424,  74256, 136136, 74256, 11424, 408,   2,   0;

G. C. Greubel, Antidiagonals n = 0..50, flattened
G. Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, 6 (1965), circa p. 82.

Cf. A006331, A006332, A006333, A006334, A040000, A132341, A154955.

Flatten[{{1}, {-1, -1}}~Join~Table[(2(-1)^(#+k)*(#+k-1)!*(2#+2k-3)!)/(#!*k!*(2# - 1)!*(2k-1)!) &@(n-k), {n,2,12}, {k,0,n}]] (* Michael De Vlieger, Mar 26 2016 *)

f=factorial
def T(n,k):
    if (k==0): return bool(n==0) - bool(n==1)
    elif (n==0): return bool(k==0) - bool(k==1)
    else: return (-1)^(n+k)*f(n+k-2)*f(2*n+2*k-2)/(f(n)*f(k)*f(2*n-1)*f(2*k-1))
flatten([[T(n-k, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Dec 14 2021

T(n, k) = (-1)^(n+k)*(n+k-2)!*(2*n+2*k-2)!/(n!*k!*(2*n-1)!*(2*k-1)!), with T(0, 0) = 1, T(0, 1) = T(1, 0) = -1.
A(n, k) = T(n-k, k) (antidiagonals).
A(n, n-k) = A(n, k).
A(2*n, n) = A132341(n).

More terms from Max Alekseyev, Sep 12 2009

Showing 1-10 of 14 results. Next

cp's OEIS Frontend

A000035 Period 2: repeat [0, 1]; a(n) = n mod 2; parity of n.

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Maxima

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A001227 Number of odd divisors of n.

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A209229 Characteristic function of powers of 2, cf. A000079.

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A108299 Triangle read by rows, 0 <= k <= n: T(n,k) = binomial(n-[(k+1)/2],[k/2])*(-1)^[(k+1)/2].

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A007331 Fourier coefficients of E_{infinity,4}.

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A132013 T(n,j) for an iterated mixed order Laguerre transform. Coefficients of the normalized generalized Laguerre polynomials (-1)^nn!L(n,1-n,x).