A134824 -id:A134824 - cp's OEIS frontend

A153881 1 followed by -1, -1, -1, ... .

1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1

Offset: 1

Mats Granvik, Jan 03 2009

Dirichlet inverse of A074206.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1).

If prefixed by initial 0, we get A134824.
Cf. A074206 (Dirichlet inverse).
Cf. A033879, A055615, A057427, A063524, A344587, A346234.

[1 - 2*Sign(n-1) : n in [1..100]]; // Wesley Ivan Hurt, Jun 20 2014

A153881:=n->`if`(n=1, 1, -1); seq(A153881(n), n=1..100); # Wesley Ivan Hurt, Jun 20 2014

Table[1 - 2 Sign[n - 1], {n, 100}] (* Wesley Ivan Hurt, Jun 20 2014 *)

a(n)=if(n>1,-1,1) \\ Charles R Greathouse IV, Sep 24 2015

G.f: x*(1-2*x)/(1-x). - Mats Granvik, Mar 09 2009, rewritten R. J. Mathar, Mar 31 2010
a(n) = (-1)^A000040(n). - Juri-Stepan Gerasimov, Sep 10 2009
G.f.: x / (1 + x / (1 - 2*x)). - Michael Somos, Apr 02 2012
From Wesley Ivan Hurt, Jun 20 2014: (Start)
a(1) = 1; a(n) = -1, n > 1.
a(n) = 1 - 2*sign(n-1) = 1 - 2*A057427(n-1).
a(n) = (-1)^sign(1-n) = (-1)^A057427(1-n).
a(n) = 2*floor(1/n)-1 = 2*A063524(n)-1. (End)
Dirichlet g.f.: 2 - zeta(s). - Álvar Ibeas, Dec 30 2018
a(n) = Sum_{d|n} A033879(d)*A055615(n/d) = Sum_{d|n} A344587(d)*A346234(n/d). - Antti Karttunen, Nov 22 2024

Edited by Charles R Greathouse IV, Mar 18 2010

More terms from Antti Karttunen, Nov 22 2024

A104967 Matrix inverse of triangle A104219, read by rows, where A104219(n,k) equals the number of Schroeder paths of length 2n having k peaks at height 1.

Original entry on oeis.org

1, -1, 1, -1, -2, 1, -1, -1, -3, 1, -1, 0, 0, -4, 1, -1, 1, 2, 2, -5, 1, -1, 2, 3, 4, 5, -6, 1, -1, 3, 3, 3, 5, 9, -7, 1, -1, 4, 2, 0, 0, 4, 14, -8, 1, -1, 5, 0, -4, -6, -6, 0, 20, -9, 1, -1, 6, -3, -8, -10, -12, -14, -8, 27, -10, 1, -1, 7, -7, -11, -10, -10, -14, -22, -21, 35, -11, 1, -1, 8, -12, -12, -5, 0, 0, -8, -27, -40, 44, -12, 1

Offset: 0

Paul D. Hanna, Mar 30 2005

Row sums equal A090132 with odd-indexed terms negated. Absolute row sums form A104968. Row sums of squared terms gives A104969.

Riordan array ((1-2*x)/(1-x), x(1-2*x)/(1-x)). - Philippe Deléham, Dec 05 2015

			Triangle begins:
   1;
  -1,  1;
  -1, -2,  1;
  -1, -1, -3,  1;
  -1,  0,  0, -4,  1;
  -1,  1,  2,  2, -5,  1;
  -1,  2,  3,  4,  5, -6,  1;
  -1,  3,  3,  3,  5,  9, -7,  1;
  -1,  4,  2,  0,  0,  4, 14, -8,  1;
  -1,  5,  0, -4, -6, -6,  0, 20, -9, 1; ...

Paul D. Hanna, Table of n, a(n) for n = 0..1080

Cf. A078011, A090132, A104968, A104969, A134824, A153881.

Cf. A347171 (rows reversed, up to signs).

A104967:= func< n,k | (&+[(-2)^j*Binomial(k+1, j)*Binomial(n-j, k): j in [0..n-k]]) >;
[A104967(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 09 2021

A104967:= (n,k)-> add( (-2)^j*binomial(k+1, j)*binomial(n-j, k), j=0..n-k);
seq(seq( A104967(n,k), k=0..n), n=0..12); # G. C. Greubel, Jun 09 2021

T[n_, k_]:= T[n, k]= Which[k==n, 1, k==0, 0, True, T[n-1, k-1] - Sum[T[n-i, k-1], {i, 2, n-k+1}]];
Table[T[n, k], {n, 13}, {k, n}]//Flatten (* Jean-François Alcover, Jun 11 2019, after Peter Luschny *)

T(n,k):=sum((-2)^i*binomial(k+1,i)*binomial(n-i,k),i,0,n-k); /* Vladimir Kruchinin, Nov 02 2011 */

{T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k)); polcoeff(polcoeff((1-2*X)/(1-X-X*Y*(1-2*X)),n,x),k,y)}
for(n=0, 16, for(k=0, n, print1(T(n, k), ", ")); print(""))

def A104967_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1,k-1)-sum(prec(n-i,k-1) for i in (2..n-k+1))
    return [prec(n, k) for k in (1..n)]
for n in (1..10): print(A104967_row(n)) # Peter Luschny, Mar 16 2016

G.f.: A(x, y) = (1-2*x)/(1-x - x*y*(1-2*x)).
Sum_{k=0..n} T(n, k) = (-1)^n*A090132(n).
Sum_{k=0..n} abs(T(n, k)) = A104968(n).
Sum_{k=0..n} T(n, k)^2 = A104969(n).
T(n,k) = Sum_{i=0..n-k} (-2)^i*binomial(k+1,i)*binomial(n-i,k). - Vladimir Kruchinin, Nov 02 2011
Sum_{k=0..floor(n/2)} T(n-k, k) = A078011(n+2). - G. C. Greubel, Jun 09 2021

cp's OEIS Frontend

A153881 1 followed by -1, -1, -1, ... .

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