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%I #22 Sep 08 2022 08:45:06
%S 1,40,1092,25664,561104,11807616,243248704,4950550528,100040447232,
%T 2013177300992,40412056994816,810023815790592,16221871691714560,
%U 324694197936160768,6496965245491888128,129976281056339296256
%N Diagonal T(n,3) of triangle in A071951.
%H G. C. Greubel, <a href="/A071952/b071952.txt">Table of n, a(n) for n = 4..250</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_04">Index entries for linear recurrences with constant coefficients</a>, signature (40, -508, 2304, -2880).
%F From _Wolfdieter Lang_, Nov 07 2003: (Start)
%F a(n+4) = A071951(n+4, 4) = (-7*2^n + 405*6^n - 2268*12^n + 2500*20^n)/630, n >= 0.
%F G.f.: x^4/((1-2*1*x)*(1-3*2*x)*(1-4*3*x)*(1-5*4*x)). (End)
%F a(n) = det(|ps(i+2,j+1)|, 1 <= i,j <= n-4), where ps(n,k) are Legendre-Stirling numbers of the first kind (A129467) and n > 3. - _Mircea Merca_, Apr 06 2013
%F From _G. C. Greubel_, Mar 16 2019: (Start)
%F a(n) = 2^(n-7)*(20*3^n - 7*6^n + 10^n - 28)/315.
%F E.g.f.: (1 - exp(2*x))^4*(14 + 28*exp(2*x) + 28*exp(4*x) + 20*exp(6*x) + 10*exp(8*x) + 4*exp(10*x) + exp(12*x))/8!. (End)
%t Flatten[ Table[ Sum[(-1)^{r + 4}(2r + 1)(r^2 + r)^n/((r + 5)!(4 - r)!), {r, 1, 4}], {n, 4, 20}]]
%t LinearRecurrence[{40, -508, 2304, -2880}, {1, 40, 1092, 25664}, 20] (* _G. C. Greubel_, Mar 16 2019 *)
%o (PARI) {a(n) = 2^(n-7)*(20*3^n - 7*6^n + 10^n - 28)/315}; \\ _G. C. Greubel_, Mar 16 2019
%o (Magma) [2^(n-7)*(20*3^n - 7*6^n + 10^n - 28)/315: n in [4..20]]; // _G. C. Greubel_, Mar 16 2019
%o (Sage) [2^(n-7)*(20*3^n - 7*6^n + 10^n - 28)/315 for n in (4..20)] # _G. C. Greubel_, Mar 16 2019
%o (GAP) List([4..20], n-> 2^(n-7)*(20*3^n - 7*6^n + 10^n - 28)/315) # _G. C. Greubel_, Mar 16 2019
%Y Cf. A000079, A016129, A015309, A089278, A089500.
%Y Cf. A071951, A071952.
%K nonn
%O 4,2
%A _N. J. A. Sloane_, Jun 16 2002
%E More terms from _Robert G. Wilson v_, Jun 19 2002
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