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A167374 Triangle, read by rows, given by [ -1,1,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. 13
1, -1, 1, 0, -1, 1, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Riordan array (1-x,1) read by rows; Riordan inverse is (1/(1-x),1). Columns have g.f. (1-x)x^k. Diagonal sums are A033999. Unsigned version in A097806.

Table T(n,k) read by antidiagonals. T(n,1) = 1, T(n,2) = -1, T(n,k) = 0, k > 2. - Boris Putievskiy, Jan 17 2013

Finite difference operator (pair difference): left multiplication by T of a sequence arranged as a column vector gives a running forward difference, a(k+1)-a(k), or first finite difference (modulo sign), of the elements of the sequence. T^n gives the n-th finite difference (mod sign). T is the inverse of the summation matrix A000012 (regarded as lower triangular matrices). - Tom Copeland, Mar 26 2014

LINKS

Table of n, a(n) for n=0..90.

Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

FORMULA

Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A011782(n), A025192(n), A002001(n), A005054(n), A052934(n), A055272(n), A055274(n), A055275(n), A055268(n), A055276(n) for x = 1,2,3,4,5,6,7,8,9,10,11 respectively .

From Boris Putievskiy, Jan 17 2013: (Start)

a(n) = floor((A002260(n)+2)/(A003056(n)+2))*(-1)^(A002260(n)+A003056(n)+1),  n>0.

a(n) = floor((i+2)/(t+2))*(-1)^(i+t+1), n > 0, where

i = n - t*(t+1)/2,

t = floor((-1 + sqrt(8*n-7))/2). (End)

T*A000012 = Identity matrix. T*A007318 = A097805. T*(A007318)^(-1)= signed A029653. - Tom Copeland, Mar 26 2014

G.f.: (1-x)/(1-x*y). - R. J. Mathar, Aug 11 2015

T = A130595*A156644 = M*T^(-1)*M = M*A000012*M, where M(n,k) = (-1)^n A130595(n,k). Note that M = M^(-1). Cf. A118800 and A097805. - Tom Copeland, Nov 15 2016

EXAMPLE

Triangle begins:

   1;

  -1,  1;

   0, -1,  1;

   0,  0, -1,  1;

   0,  0,  0, -1,  1;

   0,  0,  0,  0, -1,  1; ...

Row number r (r>4) contains (r-2) times '0', then '-1' and '1'.

From Boris Putievskiy, Jan 17 2013: (Start)

The start of the sequence as a table:

  1  -1  0  0  0  0  0 ...

  1  -1  0  0  0  0  0 ...

  1  -1  0  0  0  0  0 ...

  1  -1  0  0  0  0  0 ...

  1  -1  0  0  0  0  0 ...

  1  -1  0  0  0  0  0 ...

  1  -1  0  0  0  0  0 ...

  ...

(End)

MAPLE

A167374 := proc(n, k)

    if k> n or k < n-1 then

        0;

    elif k = n then

        1;

    else

        -1 ;

    end if;

end proc: # R. J. Mathar, Sep 07 2016

MATHEMATICA

Table[PadLeft[{-1, 1}, n], {n, 13}] // Flatten (* or *)

MapIndexed[Take[#1, First@ #2] &, CoefficientList[Series[(1 - x)/(1 - x y), {x, 0, 12}], {x, y}]] // Flatten (* Michael De Vlieger, Nov 16 2016 *)

CROSSREFS

Cf. A000012, A007318, A029653, A097805, A118800, A130595, A156644.

Sequence in context: A116938 A105589 A097806 * A132971 A085357 A011748

Adjacent sequences:  A167371 A167372 A167373 * A167375 A167376 A167377

KEYWORD

sign,tabl,easy

AUTHOR

Philippe Deléham, Nov 02 2009

STATUS

approved

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Last modified September 23 21:16 EDT 2017. Contains 292391 sequences.