A127899 - OEIS

%I #32 Jul 19 2024 11:30:00

%S 1,-2,2,0,-3,3,0,0,-4,4,0,0,0,-5,5,0,0,0,0,-6,6,0,0,0,0,0,-7,7,0,0,0,

%T 0,0,0,-8,8,0,0,0,0,0,0,0,-9,9,0,0,0,0,0,0,0,0,-10,10,0,0,0,0,0,0,0,0,

%U 0,-11,11,0,0,0,0,0,0,0,0,0,0,-12,12

%N Transform related to the harmonic series.

%C This transform is the inverse of a triangle in which each row has n terms of the harmonic series; i.e., the inverse of: 1; 1, 1/2; 1, 1/2, 1/3; ...

%C Eigensequence of the unsigned triangle = A002467 starting (1, 4, 15, 76, 455, ...). - _Gary W. Adamson_, Dec 29 2008

%C Table T(n,k) read by antidiagonals. T(1,1)=1, T(n,1) = n (for n>1), T(n,2) = -n, T(n,k) = 0, k > 2. - _Boris Putievskiy_, Jan 17 2013

%H Reinhard Zumkeller, <a href="/A127899/b127899.txt">Rows n = 1..100 of triangle, flattened</a>

%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 Triangle, a(1) = 1; by rows, (n-2) zeros followed by -n, n.

%F From _Boris Putievskiy_, Jan 17 2013: (Start)

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

%F a(n) = floor((i+2)/(t+2))*(t+1)*(-1)^(i+t+1), where i=n-t*(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). (End)

%F a(n) = floor(-1/2*A002024(n)^2 + A002024(n+1)^2-1/2*A002024(n+1) + 1/2*A002024(n+2) - 1/2*A002024(n+2)^2). - _Brian Tenneson_, Feb 10 2017

%e First few rows of the triangle are:

%e 1;

%e -2, 2;

%e 0, -3, 3;

%e 0, 0, -4, 4;

%e 0, 0, 0, -5, 5;

%e 0, 0, 0, 0, -6, 6;

%e 0, 0, 0, 0, 0, -7, 7;

%e ...

%e From _Boris Putievskiy_, Jan 17 2013: (Start)

%e The start of the sequence as table:

%e 1..-1..0..0..0..0..0...

%e 1..-2..0..0..0..0..0...

%e 2..-3..0..0..0..0..0...

%e 3..-4..0..0..0..0..0...

%e 4..-5..0..0..0..0..0...

%e 5..-6..0..0..0..0..0...

%e 6..-7..0..0..0..0..0...

%e ...

%e The start of the sequence as triangle array read by rows:

%e 1;

%e -1,1;

%e 0,-2,2;

%e 0,0,-3,3;

%e 0,0,0,-4,4;

%e 0,0,0,0,-5,5;

%e 0,0,0,0,0,-6,6;

%e 0,0,0,0,0,0,-7,7;

%e ...

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

%p A127899 := proc(n,k)

%p     if k = n then

%p         n;

%p     elif k = n-1 then

%p         -n;

%p     else

%p         0;

%p     end if;

%p end proc:

%p seq(seq( A127899(n,k),k=1..n),n=1..13) ; # _R. J. Mathar_, Jul 19 2024

%t Table[Module[{t = Floor[(-1 + Sqrt[8 n - 7])/2], i}, i = n - t (t + 1)/2; Floor[(i + 2)/(t + 2)] (t + 1) (-1)^(i + t + 1)], {n, 78}] (* or *)

%t Table[If[n == 1, {n}, ConstantArray[0, n - 2]~Join~{-n, n}], {n, 12}] // Flatten (* _Michael De Vlieger_, Feb 11 2017 *)

%o (Haskell)

%o a127899 n k = a127899_tabl !! (n-1) !! (k-1)

%o a127899_row n = a127899_tabl !! (n-1)

%o a127899_tabl = map reverse ([1] : xss) where

%o    xss = iterate (\(u : v : ws) -> u + 1 : v - 1 : ws ++ [0]) [2, -2]

%o -- _Reinhard Zumkeller_, Nov 14 2014

%Y Cf. A002467.

%K tabl,sign,easy

%O 1,2

%A _Gary W. Adamson_, Feb 04 2007