A110426 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 (see Example).

1, 3, 3, -5, -30, -84, -182, -342, -585, -935, -1419, -2067, -2912, -3990, -5340, -7004, -9027, -11457, -14345, -17745, -21714, -26312, -31602, -37650, -44525, -52299, -61047, -70847, -81780, -93930, -107384, -122232, -138567, -156485, -176085, -197469, -220742, -246012, -273390, -302990

Offset: 1

Author

Amarnath Murthy, Aug 01 2005

Keywords

• easy
• sign
• changed

Examples

			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.
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 row sums.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000330, A000914, A110425, A110427.

Programs

Mathematica

A110426[n_] := n*(n + 1)*(2 - (n - 3)*n)/8; Array[A110426, 50] (* or *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 3, 3, -5, -30}, 50] (* Paolo Xausa, Aug 26 2025 *)

PARI

Vec(x*(1 - 2*x - 2*x^2)/(1 - x)^5 + O(x^50)) \\ Colin Barker, May 27 2017

Formula

a(n) = Sum_{i=(2-n)*(n+1)/2..n} i = (-n^4 + 2*n^3 + 5*n^2 + 2*n)/8. - Theresa Guinard, Nov 15 2013

a(n) = A000330(n) - A000914(n-1). - J. M. Bergot, May 27 2017

From Colin Barker, May 27 2017: (Start)

G.f.: x*(1 - 2*x - 2*x^2)/(1 - x)^5.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5. (End)

E.g.f.: -x*(-8 - 4*x + 4*x^2 + x^3)*exp(x)/8. - Elmo R. Oliveira, Aug 24 2025

Extensions

More terms from Joshua Zucker, May 10 2006