A111967 -id:A111967 - cp's OEIS frontend

A111982 Row sums of abs(A111967).

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

Offset: 0

Paul Barry, Aug 24 2005

mod(a(n),2)=A036987(n). a(2^n-1)=2n-1+2*0^n; a(2^n)=2-0^n.

A127749 Inverse of number triangle A(n,k) = 1/(2n+1) if k <= n <= 2k, 0 otherwise.

Original entry on oeis.org

1, 0, 3, 0, -3, 5, 0, 3, -5, 7, 0, 0, 0, -7, 9, 0, -3, 5, 0, -9, 11, 0, 0, 0, 0, 0, -11, 13, 0, 3, -5, 7, 0, 0, -13, 15, 0, 0, 0, 0, 0, 0, 0, -15, 17, 0, 0, 0, -7, 9, 0, 0, 0, -17, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, -19, 21, 0, -3, 5

Offset: 0

Paul Barry, Jan 28 2007

Conjectures: row sums modulo 2 are the Fredholm-Rueppel sequence A036987; row sums of triangle modulo 2 are A111982. Row sums are A127750.

The first conjecture is equivalent to the row sums conjecture in A111967. - R. J. Mathar, Apr 21 2021

			Triangle begins
  1;
  0,  3;
  0, -3,  5;
  0,  3, -5,  7;
  0,  0,  0, -7,  9;
  0, -3,  5,  0, -9,  11;
  0,  0,  0,  0,  0, -11,  13;
  0,  3, -5,  7,  0,   0, -13,  15;
  0,  0,  0,  0,  0,   0,   0, -15,  17;
  0,  0,  0, -7,  9,   0,   0,   0, -17,  19;
  0,  0,  0,  0,  0,   0,   0,   0,   0, -19,  21;
  0, -3,  5,  0, -9,  11,   0,   0,   0,   0, -21,  23;
  0,  0,  0,  0,  0,   0,   0,   0,   0,   0,   0, -23, 25;
Inverse of triangle
  1;
  0, 1/3;
  0, 1/5, 1/5;
  0,  0,  1/7, 1/7;
  0,  0,  1/9, 1/9,  1/9;
  0,  0,   0,  1/11, 1/11, 1/11;
  0,  0,   0,  1/13, 1/13, 1/13, 1/13;
  0,  0,   0,   0,   1/15, 1/15, 1/15, 1/15;
  0,  0,   0,   0,   1/17, 1/17, 1/17, 1/17, 1/17;
  0,  0,   0,   0,    0,   1/19, 1/19, 1/19, 1/19, 1/19;
  0,  0,   0,   0,    0,   1/21, 1/21, 1/21, 1/21, 1/21, 1/21;

Cf. A111967.

A127749 := proc(n,k)
    option remember ;
    if k > n then
        0 ;
    elif k = n then
        2*n+1 ;
    else
        -(2*k+1)*add( procname(n,i)/(2*i+1),i=k+1..min(n,2*k)) ;
    end if;
end proc:
seq(seq( A127749(n,k),k=0..n),n=0..20) ; # R. J. Mathar, Feb 09 2021

nmax = 10;
A[n_, k_] := If[k <= n <= 2k, 1/(2n+1), 0];
invA = Inverse[Table[A[n, k], {n, 0, nmax}, {k, 0, nmax}]];
T[n_, k_] := invA[[n+1, k+1]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 05 2020 *)

T(n,k) = (2*k+1)*A111967(n,k). - R. J. Mathar, Apr 21 2021

A127822 Triangle whose row sums modulo 2 give the Fredholm-Rueppel sequence A036987.

Original entry on oeis.org

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

Offset: 0

Paul Barry, Jan 30 2007

Row sums are A111982. Unsigned version of A111967.

			Triangle begins
1,
0, 1,
0, 1, 1,
0, 1, 1, 1,
0, 0, 0, 1, 1,
0, 1, 1, 0, 1, 1,
0, 0, 0, 0, 0, 1, 1,
0, 1, 1, 1, 0, 0, 1, 1,
0, 0, 0, 0, 0, 0, 0, 1, 1,
0, 0, 0, 1, 1, 0, 0, 0, 1, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1,
0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

G.f. of k-th column is x^k*if(k=0,1,x*sum{j=0..\infty, x^(-2^(j/2)*(((k+2)/(2*sqrt(2))-(k+1))(-1)^j-(k+2)/(2*sqrt(2))-(k+1))-(k+2))+1+x}

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A111982 Row sums of abs(A111967).

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A127749 Inverse of number triangle A(n,k) = 1/(2n+1) if k <= n <= 2k, 0 otherwise.

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A127822 Triangle whose row sums modulo 2 give the Fredholm-Rueppel sequence A036987.

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