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A071952 Diagonal T(n, 4) of triangle in A071951.

%I A071952 #35 Jul 08 2025 11:03:53
%S A071952 1,40,1092,25664,561104,11807616,243248704,4950550528,100040447232,
%T A071952 2013177300992,40412056994816,810023815790592,16221871691714560,
%U A071952 324694197936160768,6496965245491888128,129976281056339296256
%N A071952 Diagonal T(n, 4) of triangle in A071951.
%H A071952 G. C. Greubel, <a href="/A071952/b071952.txt">Table of n, a(n) for n = 4..250</a>
%H A071952 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 A071952 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 A071952 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (40,-508,2304,-2880).
%F A071952 From _Wolfdieter Lang_, Nov 07 2003: (Start)
%F A071952 a(n+4) = A071951(n+4, 4) = (-7*2^n + 405*6^n - 2268*12^n + 2500*20^n)/630, n >= 0.
%F A071952 G.f.: x^4/((1-2*1*x)*(1-3*2*x)*(1-4*3*x)*(1-5*4*x)). (End)
%F A071952 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 A071952 From _G. C. Greubel_, Mar 16 2019: (Start)
%F A071952 a(n) = 2^(n-7)*(20*3^n - 7*6^n + 10^n - 28)/315.
%F A071952 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 A071952 Flatten[ Table[ Sum[(-1)^{r + 4}(2r + 1)(r^2 + r)^n/((r + 5)!(4 - r)!), {r, 1, 4}], {n, 4, 20}]]
%t A071952 LinearRecurrence[{40, -508, 2304, -2880}, {1, 40, 1092, 25664}, 20] (* _G. C. Greubel_, Mar 16 2019 *)
%o A071952 (PARI) {a(n) = 2^(n-7)*(20*3^n - 7*6^n + 10^n - 28)/315}; \\ _G. C. Greubel_, Mar 16 2019
%o A071952 (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 A071952 (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 A071952 (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 A071952 Cf. A000079, A016129, A015309, A089278, A089500.
%Y A071952 Cf. A071951, A129467.
%K A071952 nonn,easy
%O A071952 4,2
%A A071952 _N. J. A. Sloane_, Jun 16 2002
%E A071952 More terms from _Robert G. Wilson v_, Jun 19 2002
%E A071952 Definition corrected by _Georg Fischer_, Jul 07 2025