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A187756 a(n) = n^2 * (4*n^2 - 1) / 3.

%I A187756 #36 Oct 21 2022 21:11:01
%S A187756 0,1,20,105,336,825,1716,3185,5440,8721,13300,19481,27600,38025,51156,
%T A187756 67425,87296,111265,139860,173641,213200,259161,312180,372945,442176,
%U A187756 520625,609076,708345,819280,942761,1079700,1231041,1397760,1580865,1781396,2000425
%N A187756 a(n) = n^2 * (4*n^2 - 1) / 3.
%H A187756 G. C. Greubel, <a href="/A187756/b187756.txt">Table of n, a(n) for n = 0..5000</a>
%H A187756 P. Aluffi, <a href="https://arxiv.org/abs/1408.1702">Degrees of projections of rank loci</a>, arXiv:1408.1702 [math.AG], 2014. ["After compiling the results of many explicit computations, we noticed that many of the numbers d_{n,r,S} appear in the existing literature in contexts far removed from the enumerative geometry of rank conditions; we owe this surprising (to us) observation to perusal of [Slo14]."]
%H A187756 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5, -10, 10, -5, 1).
%F A187756 G.f.: x * (1 + x) * (1 + 14*x + x^2) / (1 - x)^5.
%F A187756 a(n) = a(-n) for all n in Z.
%F A187756 a(n) = n * A000447(n).
%F A187756 G.f. A144853(x) = 1 / (1 - a(1)*x / (1 - a(2)*x / (1 - a(3)*x / ... ))).
%e A187756 G.f. = x + 20*x^2 + 105*x^3 + 336*x^4 + 825*x^5 + 1716*x^6 + 3185*x^7 + ...
%t A187756 LinearRecurrence[{5,-10,10,-5,1},{0,1,20,105,336},40] (* _Harvey P. Dale_, Mar 26 2016 *)
%t A187756 a[ n_] := SeriesCoefficient[ x * (1 + x) * (1 + 14*x + x^2) / (1 - x)^5, {x, 0, Abs[n]}]; (* _Michael Somos_, Dec 26 2016 *)
%o A187756 (PARI) {a(n) = polcoeff( x * (1 + x) * (1 + 14*x + x^2) / (1 - x)^5 + x * O(x^n), abs(n))};
%o A187756 (Maxima) A187756(n):=n^2*(4*n^2-1)/3$ makelist(A187756(n),n,0,20); /* _Martin Ettl_, Jan 07 2013 */
%o A187756 (Magma) [n^2*(4*n^2-1)/3: n in [0..50]]; // _G. C. Greubel_, Aug 10 2018
%Y A187756 Cf. A000447, A144853.
%K A187756 nonn,easy
%O A187756 0,3
%A A187756 _Michael Somos_, Jan 03 2013