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%I #47 Oct 01 2022 23:23:50
%S 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,
%T 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,
%U -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
%N 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.
%C 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.
%C 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
%C 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
%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations [of] Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.
%F 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 .
%F From _Boris Putievskiy_, Jan 17 2013: (Start)
%F a(n) = floor((A002260(n)+2)/(A003056(n)+2))*(-1)^(A002260(n)+A003056(n)+1), n>0.
%F a(n) = floor((i+2)/(t+2))*(-1)^(i+t+1), n > 0, where
%F i = n - t*(t+1)/2,
%F t = floor((-1 + sqrt(8*n-7))/2). (End)
%F T*A000012 = Identity matrix. T*A007318 = A097805. T*(A007318)^(-1)= signed A029653. - _Tom Copeland_, Mar 26 2014
%F G.f.: (1-x)/(1-x*y). - _R. J. Mathar_, Aug 11 2015
%F 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
%e Triangle begins:
%e 1;
%e -1, 1;
%e 0, -1, 1;
%e 0, 0, -1, 1;
%e 0, 0, 0, -1, 1;
%e 0, 0, 0, 0, -1, 1; ...
%e Row number r (r>4) contains (r-2) times '0', then '-1' and '1'.
%e From _Boris Putievskiy_, Jan 17 2013: (Start)
%e The start of the sequence as a table:
%e 1 -1 0 0 0 0 0 ...
%e 1 -1 0 0 0 0 0 ...
%e 1 -1 0 0 0 0 0 ...
%e 1 -1 0 0 0 0 0 ...
%e 1 -1 0 0 0 0 0 ...
%e 1 -1 0 0 0 0 0 ...
%e 1 -1 0 0 0 0 0 ...
%e ...
%e (End)
%p A167374 := proc(n,k)
%p if k> n or k < n-1 then
%p 0;
%p elif k = n then
%p 1;
%p else
%p -1 ;
%p end if;
%p end proc: # _R. J. Mathar_, Sep 07 2016
%t Table[PadLeft[{-1, 1}, n], {n, 13}] // Flatten (* or *)
%t MapIndexed[Take[#1, First@ #2] &, CoefficientList[Series[(1 - x)/(1 - x y), {x, 0, 12}], {x, y}]] // Flatten (* _Michael De Vlieger_, Nov 16 2016 *)
%t T[n_, k_] := If[ k<0 || k>n, 0, Boole[n==k] - Boole[n==k+1]]; (* _Michael Somos_, Oct 01 2022 *)
%o (PARI) {T(n, k) = if( k<0 || k>n, 0, (n==k) - (n==k+1))}; /* _Michael Somos_, Oct 01 2022 */
%Y Cf. A000012, A007318, A029653, A097805, A118800, A130595, A156644.
%K sign,tabl,easy
%O 0,1
%A _Philippe Deléham_, Nov 02 2009
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