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A128624 Row sums of A128623.

%I A128624 #19 Mar 12 2024 08:56:12
%S A128624 1,4,12,24,45,72,112,160,225,300,396,504,637,784,960,1152,1377,1620,
%T A128624 1900,2200,2541,2904,3312,3744,4225,4732,5292,5880,6525,7200,7936,
%U A128624 8704,9537,10404,11340,12312,13357,14440,15600,16800,18081,19404,20812,22264,23805
%N A128624 Row sums of A128623.
%C A128624 Also the number of (w,x,y) with all terms in {0,...,n-1} and w <= R <= x, where R = max(w,x,y)-min(w,x,y), see A212959. - _Clark Kimberling_, Jun 10 2012
%H A128624 Colin Barker, <a href="/A128624/b128624.txt">Table of n, a(n) for n = 1..1000</a>
%H A128624 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).
%F A128624 G.f.: x*(1+2*x+3*x^2) / ((1+x)^2*(1-x)^4). - _R. J. Mathar_, Jun 27 2012
%F A128624 From _Colin Barker_, Jan 31 2016: (Start)
%F A128624 a(n) = n*(2*n^2 + 4*n + 1 - (-1)^n)/8.
%F A128624 a(n) = n^2*(n + 2)/4 for n even.
%F A128624 a(n) = n*(n^2 + 2*n + 1)/4 for n odd. (End)
%F A128624 From _G. C. Greubel_, Mar 12 2024: (Start)
%F A128624 a(n) = Sum_{k=0..floor((n-1)/2)} A094728(n, k).
%F A128624 E.g.f.: (1/8)*x*(exp(-x) + (7 + 10*x + 2*x^2)*exp(x)). (End)
%t A128624 Table[n*(n^2 +2*n +Mod[n,2])/4, {n,50}] (* _G. C. Greubel_, Mar 12 2024 *)
%o A128624 (PARI) Vec(x*(1+2*x+3*x^2)/((1-x)^4*(1+x)^2) + O(x^100)) \\ _Colin Barker_, Jan 31 2016
%o A128624 (Magma) [n*((n+1)^2-1+(n mod 2))/4: n in [1..50]]; // _G. C. Greubel_, Mar 12 2024
%o A128624 (SageMath) [n*((n+1)^2-1+(n%2))//4 for n in range(1,51)] # _G. C. Greubel_, Mar 12 2024
%Y A128624 Cf. A128621, A128623, A212959.
%Y A128624 Cf. A094728 (diagonal row sums).
%K A128624 nonn,easy
%O A128624 1,2
%A A128624 _Gary W. Adamson_, Mar 14 2007
%E A128624 Incorrect formula removed by _R. J. Mathar_, Jun 27 2012