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%I #70 Oct 03 2022 11:59:33
%S 1,20,117,400,1025,2196,4165,7232,11745,18100,26741,38160,52897,71540,
%T 94725,123136,157505,198612,247285,304400,370881,447700,535877,636480,
%U 750625,879476,1024245,1186192,1366625,1566900,1788421,2032640,2301057,2595220,2916725,3267216,3648385
%N The sum of the numbers of the central diamond of the multiplication table [1..k] X [1..k] for k=2*n-1.
%C a(n) is the sum of the elements of the multiplication table, forming the maximum diamond in its center.
%H Nicolay Avilov, <a href="/A357042/a357042.jpg">Drawing for a(1)-a(5)</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = n^2*(2*n^2 - 2*n + 1).
%F а(n) = 2*A000583(n) - A015237(n).
%F From _Stefano Spezia_, Sep 19 2022: (Start)
%F G.f.: x*(1 + 15*x + 27*x^2 + 5*x^3)/(1 - x)^5.
%F a(n) = A000290(n)*A001844(n-1). (End)
%e In the multiplication table [1..3] X [1..3]: a(2) = 2+2+4+6+6 = 20;
%e In the multiplication table [1..5] X [1..5]: a(3) = 3+4+3+6+6+8+9+8+12+12+15+16+15 = 117.
%e For n=3, the multiplication table [1..5] X [1..5] and the terms summed are
%e * 1 2 3 4 5
%e -----------------
%e 1| 3
%e 2| 4 6 8
%e 3| 3 6 9 12 15
%e 4| 8 12 16
%e 5| 15
%Y Cf. A000290, A001844, A003991.
%Y Cf. A000583, A015237.
%K nonn,easy
%O 1,2
%A _Nicolay Avilov_, Sep 18 2022
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