A115514 Triangle read by rows: row n >= 1 lists first n positive terms of A004526 (integers repeated) in decreasing order.

1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 1, 3, 3, 2, 2, 1, 1, 4, 3, 3, 2, 2, 1, 1, 4, 4, 3, 3, 2, 2, 1, 1, 5, 4, 4, 3, 3, 2, 2, 1, 1, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 6, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 7, 6, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 7, 7, 6, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1

Offset: 1

Author

Roger L. Bagula, Mar 07 2006

Keywords

• nonn
• tabl

Comments

T(n,k) = number of 2-element subsets of {1,2,...,n+2} such that the absolute difference of the elements is k+1, where 1 <= k < = n. E.g., T(7,3) = 3, the subsets are {1,5}, {2,6}, and {3,7}. - Christian Barrientos, Jun 27 2022

Examples

			Triangle begins as, for n >= 1, 1 <= k <= n,
  1;
  1, 1;
  2, 1, 1;
  2, 2, 1, 1;
  3, 2, 2, 1, 1;
  3, 3, 2, 2, 1, 1;
  4, 3, 3, 2, 2, 1, 1;
  ...
		

Links

• G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Crossrefs

Cf. A000012, A000073, A004526, A008805, A008967.

Cf. A002620 (row sums), A008805 (diagonal sums), A142150 (alternating row sums)

Programs

Magma

[Floor((n-k+2)/2): k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 14 2024

Maple

# Assuming offset 0:
Even := n -> (1 + (-1)^n)/2: # Iverson's even.
p := n -> add(add(Even(k)*x^j, j = 0..n-k), k = 0..n):
for n from 0 to 9 do seq(coeff(p(n), x, k), k=0..n) od; # Peter Luschny, Jun 03 2021

Mathematica

Table[Floor[(n-k+2)/2], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 14 2024 *)

SageMath

flatten([[(n-k+2)//2 for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 14 2024

Formula

Sum_{k=1..n} T(n, k) = A002620(n+1) (row sums). - Gary W. Adamson, Oct 25 2007

T(n, k) = [x^k] p(n), where p(n) are partial Gaussian polynomials (A008967) defined by p(n) = Sum_{k=0..n} Sum_{j=0..n-k} even(k)*x^j, and even(k) = 1 if k is even and otherwise 0. We assume offset 0. - Peter Luschny, Jun 03 2021

T(n, k) = floor((n+2-k)/2). - Christian Barrientos, Jun 27 2022

From G. C. Greubel, Mar 14 2024: (Start)

T(n, k) = A128623(n, k)/n.

Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A142150(n+1).

Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A008805(n-1).

Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = A002265(n+3). (End)

Extensions

Edited by N. J. A. Sloane, Mar 23 2008 and Dec 15 2017