A115361 -id:A115361 - cp's OEIS frontend

A129265 Triangle read by rows: T(n,k) is the number of power of two divisors of n that are less than or equal to n/k.

1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Apr 06 2007

Equals A115361 * A000012 as infinite lower triangular matrices (cf. A129264).

			First few rows of the triangle are:
  1;
  2, 1;
  1, 1, 1;
  3, 2, 1, 1;
  1, 1, 1, 1, 1;
  2, 2, 2, 1, 1, 1;
  1, 1, 1, 1, 1, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Row sums are A129527.
Column 1 is A001511.
Cf. A115361, A000012, A129264.

T(n, k)={sumdiv(n, d, d <= n/k && d==1<Andrew Howroyd, Aug 07 2018

T(n,k) = 1 for n odd.

Name changed and terms a(56) and beyond from Andrew Howroyd, Aug 07 2018

A129503 Pascal's Fredholm-Rueppel triangle.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 3, 0, 1, 1, 4, 0, 3, 0, 1, 5, 0, 6, 0, 0, 1, 6, 0, 10, 0, 0, 0, 1, 7, 0, 15, 0, 0, 0, 1, 1, 8, 0, 21, 0, 0, 0, 4, 0, 1, 9, 0, 28, 0, 0, 0, 10, 0, 0, 1, 10, 0, 36, 0, 0, 0, 20, 0, 0, 0, 1, 11, 0, 45, 0, 0, 0, 35, 0, 0, 0, 0, 1, 12, 0, 55, 0, 0, 0, 56, 0, 0, 0, 0, 0

Offset: 1

Gary W. Adamson, Apr 18 2007

First row of the array = the Fredholm-Rueppel sequence (A036987); which becomes the right border of the triangle. Second row of the array (1, 2, 0, 3, 0, 0, 0, 4, ...) = A104117. Third row of the array (1, 3, 0, 6, 0, 0, 0, 10, ...) = A129502. Row sums of triangle A129503 = A129504: (1, 2, 3, 5, 8, 12, 17, 24, 34, ...).

			First few rows of the triangle:
  1;
  1,  1;
  1,  2,  0;
  1,  3,  0,  1;
  1,  4,  0,  3,  0;
  1,  5,  0,  6,  0,  0;
  1,  6,  0, 10,  0,  0,  0;
  1,  7,  0, 15,  0,  0,  0,  1;
  1,  8,  0, 21,  0,  0,  0,  4,  0;
  1,  9,  0, 28,  0,  0,  0, 10,  0,  0;
  1, 10,  0, 36,  0,  0,  0, 20,  0,  0,  0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Row sums are A129504.

Cf. A036987, A115361, A104117, A129502.

T(n,k)=my(e=valuation(k,2)); if(k==2^e, binomial(n-k+e, e)) \\ Andrew Howroyd, Aug 09 2018

Antidiagonals of an array in which n-th row (n=0,1,2,...) = M^n * V, where M = A115361 as an infinite lower triangular matrix and V = the Fredholm-Rueppel sequence A036987 as a vector: [1, 1, 0, 1, 0, 0, 0, 1, ...]. The array = 1, 1, 0, 1, 0, 0, 0, 1, 0, ... 1, 2, 0, 3, 0, 0, 0, 4, 0, ... 1, 3, 0, 6, 0, 0, 0, 10, 0, ... 1, 4, 0, 10, 0, 0, 0, 20, 0, ... (n+1)-th row can be generated from A115361 * n-th row.

T(n, 2^e) = binomial(n + e - 2^e, e), T(n, k) = 0 otherwise. - Andrew Howroyd, Aug 09 2018

a(53) corrected and terms a(67) and beyond from Andrew Howroyd, Aug 09 2018

A115363 ((1,x)-(x,x^2))^(-2) (using Riordan array notation).

Original entry on oeis.org

1, 2, 1, 0, 0, 1, 3, 2, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 4, 3, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 3, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Paul Barry, Jan 21 2006

Square of number triangle A115361. Row sums are A115364.

			Triangle begins
1,
2, 1,
0, 0, 1,
3, 2, 0, 1,
0, 0, 0, 0, 1,
0, 0, 2, 0, 0, 1,
0, 0, 0, 0, 0, 0, 1,
4, 3, 0, 2, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 2, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 3, 0, 0, 2, 0, 0, 0, 0, 0, 1,

cp's OEIS Frontend

A129265 Triangle read by rows: T(n,k) is the number of power of two divisors of n that are less than or equal to n/k.

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A129503 Pascal's Fredholm-Rueppel triangle.

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PARI

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Extensions

A115363 ((1,x)-(x,x^2))^(-2) (using Riordan array notation).

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Author

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Examples