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%I #23 Sep 08 2022 08:45:06
%S 4,52,292,1092,3192,7896,17304,34584,64284,112684,188188,301756,
%T 467376,702576,1028976,1472880,2065908,2845668,3856468,5150068,
%U 6786472,8834760,11373960,14493960,18296460,22895964,28420812,35014252
%N Diagonal T(n,n-2) of triangle in A071951.
%H G. C. Greubel, <a href="/A071953/b071953.txt">Table of n, a(n) for n = 3..5000</a>
%H W. N. Everitt, L. L. Littlejohn and R. Wellman, <a href="http://dx.doi.org/10.1016/S0377-0427(02)00582-4">Legendre polynomials, Legendre-Stirling numbers and the left-definite spectral analysis of the Legendre differential expression</a>, J. Comput. Appl. Math. 148, 2002, 213-238.
%H L. L. Littlejohn and R. Wellman, <a href="http://dx.doi.org/10.1006/jdeq.2001.4078">A general left-definite theory for certain self-adjoint operators with applications to differential equations</a>, J. Differential Equations, 181(2), 2002, 280-339.
%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 a(n) = (n-2)*(n-1)*n*(n+1)*(5*n^2 - 11*n + 3)/90.
%F a(0)=4, a(1)=52, a(2)=292, a(3)=1092, a(4)=3192, a(5)=7896, a(6)=17304, 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). - _Harvey P. Dale_, Jul 03 2011
%F G.f.: 4*(3*x*(x+2)+1)/(1-x)^7. - _Harvey P. Dale_, Jul 03 2011
%F E.g.f.: x^3*(60 + 135*x + 54*x^2 + 5*x^3)*exp(x)/90. - _G. C. Greubel_, Mar 16 2019
%t Flatten[ Table[ Sum[(-1)^{r + n - 2}(2r + 1)(r^2 + r)^n/((r + n - 1)!(n - 2 - r)!), {r, 1, n - 2}], {n, 3, 34}]]
%t Table[(n-2)(n-1)n(n+1)(5n^2-11n+3)/90,{n,3,30}] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{4,52,292,1092,3192, 7896,17304}, 30] (* _Harvey P. Dale_, Jul 03 2011 *)
%o (PARI) {a(n) = (n-2)*(n-1)*n*(n+1)*(5*n^2 - 11*n + 3)/90}; \\ _G. C. Greubel_, Mar 16 2019
%o (Magma) [(n-2)*(n-1)*n*(n+1)*(5*n^2 - 11*n + 3)/90: n in [3..30]]; // _G. C. Greubel_, Mar 16 2019
%o (Sage) [(n-2)*(n-1)*n*(n+1)*(5*n^2 - 11*n + 3)/90 for n in (3..30)] # _G. C. Greubel_, Mar 16 2019
%o (GAP) List([3..30], n-> (n-2)*(n-1)*n*(n+1)*(5*n^2 - 11*n + 3)/90); # _G. C. Greubel_, Mar 16 2019
%Y Cf. A071951, A071952.
%K nonn
%O 3,1
%A _N. J. A. Sloane_, Jun 16 2002
%E More terms from _Robert G. Wilson v_, Jun 19 2002
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