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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.

%I A110427 #34 Aug 26 2025 12:41:52
%S A110427 1,1,-3,-14,-35,-69,-119,-188,-279,-395,-539,-714,-923,-1169,-1455,
%T A110427 -1784,-2159,-2583,-3059,-3590,-4179,-4829,-5543,-6324,-7175,-8099,
%U A110427 -9099,-10178,-11339,-12585,-13919,-15344,-16863,-18479,-20195,-22014,-23939,-25973,-28119,-30380,-32759,-35259,-37883
%N 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.
%H A110427 Paolo Xausa, <a href="/A110427/b110427.txt">Table of n, a(n) for n = 1..10000</a>
%H A110427 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A110427 From _R. J. Mathar_, Jul 10 2009: (Start)
%F A110427 a(n) = n*(1 + 2*n - n^2)/2 = n - A002411(n-1).
%F A110427 G.f.: x*(1 - 3*x - x^2)/(1-x)^4.
%F A110427 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)
%F A110427 E.g.f.: -x*(-1 + x)*(2 + x)*exp(x)/2. - _Elmo R. Oliveira_, Aug 24 2025
%e A110427 E.g., the row corresponding to 4 contains 4, (3+2),{(1) +(0)+(-1)}, {(-2)+(-3)+(-4)+(-5)} ----> 4,5,0,-14
%e A110427   1
%e A110427   2  1
%e A110427   3  3 -3
%e A110427   4  5  0 -14
%e A110427   5  7  3 -10 -35
%e A110427   6  9  6  -6 -30 -69
%e A110427   ...
%e A110427 Sequence contains the diagonal.
%t A110427 A110427[n_] := n*(1 - (n - 2)*n)/2; Array[A110427, 50] (* or *)
%t A110427 LinearRecurrence[{4, -6, 4, -1}, {1, 1, -3, -14}, 50] (* _Paolo Xausa_, Aug 25 2025 *)
%o A110427 (PLT Scheme)
%o A110427 (first (reverse (A110425 n))
%o A110427 ;;see A110425 for definition of that function.
%o A110427 -- _Joshua Zucker_, May 10 2006
%Y A110427 Cf. A110425, A110426.
%K A110427 easy,sign,changed
%O A110427 1,3
%A A110427 _Amarnath Murthy_, Aug 01 2005
%E A110427 More terms from _Joshua Zucker_, May 10 2006