A110427 The r-th term of the n-th row of the following array contains the sum of r successively decreasing integers beginning from n. 0 < r <= n. Sequence contains the leading diagonal.

1, 1, -3, -14, -35, -69, -119, -188, -279, -395, -539, -714, -923, -1169, -1455, -1784, -2159, -2583, -3059, -3590, -4179, -4829, -5543, -6324, -7175, -8099, -9099, -10178, -11339, -12585, -13919, -15344, -16863, -18479, -20195, -22014, -23939, -25973, -28119, -30380, -32759, -35259, -37883

Offset: 1

Author

Amarnath Murthy, Aug 01 2005

Keywords

• easy
• sign
• changed

Examples

			E.g., the row corresponding to 4 contains 4, (3+2),{(1) +(0)+(-1)}, {(-2)+(-3)+(-4)+(-5)} ----> 4,5,0,-14
  1
  2  1
  3  3 -3
  4  5  0 -14
  5  7  3 -10 -35
  6  9  6  -6 -30 -69
  ...
Sequence contains the diagonal.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A110425, A110426.

Programs

Mathematica

A110427[n_] := n*(1 - (n - 2)*n)/2; Array[A110427, 50] (* or *)
LinearRecurrence[{4, -6, 4, -1}, {1, 1, -3, -14}, 50] (* Paolo Xausa, Aug 25 2025 *)

Formula

From R. J. Mathar, Jul 10 2009: (Start)

a(n) = n*(1 + 2*n - n^2)/2 = n - A002411(n-1).

G.f.: x*(1 - 3*x - x^2)/(1-x)^4.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

E.g.f.: -x*(-1 + x)*(2 + x)*exp(x)/2. - Elmo R. Oliveira, Aug 24 2025

Extensions

More terms from Joshua Zucker, May 10 2006