A110425 -id:A110425 - cp's OEIS frontend

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

Amarnath Murthy, Aug 01 2005

			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.

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

Cf. A110425, A110426.

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 *)

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

More terms from Joshua Zucker, May 10 2006

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).

Original entry on oeis.org

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

Amarnath Murthy, Aug 01 2005

			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.

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).

Cf. A000330, A000914, A110425, A110427.

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 *)

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

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

More terms from Joshua Zucker, May 10 2006

cp's OEIS Frontend

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.

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