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a(n) = (n^6 - 30*n^4 + 45*n^3 + 206*n^2 - 576*n + 384)/6.
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%I #27 Sep 08 2022 08:45:32

%S 0,4,112,859,3640,11250,28544,63217,126704,235200,410800,682759,

%T 1088872,1676974,2506560,3650525,5197024,7251452,9938544,13404595,

%U 17819800,23380714,30312832,38873289,49353680,62083000,77430704,95809887,117680584,143553190

%N a(n) = (n^6 - 30*n^4 + 45*n^3 + 206*n^2 - 576*n + 384)/6.

%H Vincenzo Librandi, <a href="/A135917/b135917.txt">Table of n, a(n) for n = 2..1000</a>

%H Geir Ellingsrud and Stein Arild Strømme, <a href="https://arxiv.org/abs/alg-geom/9411005">Bott's formula and enumerative geometry</a>, arXiv:alg-geom/9411005, 1994.

%H Geir Ellingsrud and Stein Arild Strømme, <a href="https://doi.org/10.1090/S0894-0347-96-00189-0">Bott's formula and enumerative geometry</a>, J. Amer. Math. Soc. 9 (1996), 175-193.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1)

%F G.f.: x^3*(-4 - 84*x - 159*x^2 + 161*x^3 - 29*x^4 - 5*x^5) / (x-1)^7. - _Harvey P. Dale_, Apr 23 2011

%F a(n) = 7*a(n - 1) - 21*a(n - 2) + 35*a(n - 3) - 35*a(n - 4) + 21*a(n - 5) - 7*a(n - 6) + a(n - 7) for n > 9. - _Stefano Spezia_, Sep 03 2018

%t Table[(n^6-30n^4+45n^3+206n^2-576n+384)/6,{n,2,40}] (* or *) CoefficientList[Series[(-4x-84x^2-159x^3+161x^4-29x^5-5x^6)/ (x-1)^7,{x,0,40}],x] (* _Harvey P. Dale_, Apr 23 2011 *)

%t LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 4, 112, 859, 3640, 11250, 28544}, 40] (* _Stefano Spezia_, Sep 03 2018 *)

%o (Magma) [(n^6 - 30*n^4 + 45*n^3 + 206*n^2 - 576*n + 384)/6: n in [2..35]]; // _Vincenzo Librandi_, May 04 2011

%o (PARI) a(n) = (n^6 - 30*n^4 + 45*n^3 + 206*n^2 - 576*n + 384)/6; \\ _Andrew Howroyd_, Nov 06 2018

%K nonn,easy

%O 2,2

%A _N. J. A. Sloane_, Mar 07 2008