cp's OEIS Frontend

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).

%I A110426 #27 Aug 26 2025 08:49:20
%S A110426 1,3,3,-5,-30,-84,-182,-342,-585,-935,-1419,-2067,-2912,-3990,-5340,
%T A110426 -7004,-9027,-11457,-14345,-17745,-21714,-26312,-31602,-37650,-44525,
%U A110426 -52299,-61047,-70847,-81780,-93930,-107384,-122232,-138567,-156485,-176085,-197469,-220742,-246012,-273390,-302990
%N 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).
%H A110426 Colin Barker, <a href="/A110426/b110426.txt">Table of n, a(n) for n = 1..1000</a>
%H A110426 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A110426 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
%F A110426 a(n) = A000330(n) - A000914(n-1). - _J. M. Bergot_, May 27 2017
%F A110426 From _Colin Barker_, May 27 2017: (Start)
%F A110426 G.f.: x*(1 - 2*x - 2*x^2)/(1 - x)^5.
%F A110426 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)
%F A110426 E.g.f.: -x*(-8 - 4*x + 4*x^2 + x^3)*exp(x)/8. - _Elmo R. Oliveira_, Aug 24 2025
%e 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.
%e A110426 E.g., the row corresponding to 4 contains 4, (3+2),{(1) +(0)+(-1)}, {(-2)+(-3)+(-4)+(-5)} ----> 4,5,0,-14
%e A110426   1
%e A110426   2 1
%e A110426   3 3 -3
%e A110426   4 5 0 -14
%e A110426   5 7 3 -10 -35
%e A110426   6 9 6 -6 -30 -69
%e A110426   ...
%e A110426 Sequence contains the row sums.
%t A110426 A110426[n_] := n*(n + 1)*(2 - (n - 3)*n)/8; Array[A110426, 50] (* or *)
%t A110426 LinearRecurrence[{5, -10, 10, -5, 1}, {1, 3, 3, -5, -30}, 50] (* _Paolo Xausa_, Aug 26 2025 *)
%o A110426 ;;PLT DrScheme (Zucker)
%o A110426 (apply + (A110425 n))
%o A110426 ;;see A110425 for definition of that function.
%o A110426 (PARI) Vec(x*(1 - 2*x - 2*x^2)/(1 - x)^5 + O(x^50)) \\ _Colin Barker_, May 27 2017
%Y A110426 Cf. A000330, A000914, A110425, A110427.
%K A110426 easy,sign,changed
%O A110426 1,2
%A A110426 _Amarnath Murthy_, Aug 01 2005
%E A110426 More terms from _Joshua Zucker_, May 10 2006