A071952 Diagonal T(n, 4) of triangle in A071951.
1, 40, 1092, 25664, 561104, 11807616, 243248704, 4950550528, 100040447232, 2013177300992, 40412056994816, 810023815790592, 16221871691714560, 324694197936160768, 6496965245491888128, 129976281056339296256
Offset: 4
Links
- G. C. Greubel, Table of n, a(n) for n = 4..250
- W. N. Everitt, L. L. Littlejohn and R. Wellman, Legendre polynomials, Legendre-Stirling numbers and the left-definite spectral analysis of the Legendre differential expression, J. Comput. Appl. Math. 148, 2002, 213-238.
- L. L. Littlejohn and R. Wellman, A general left-definite theory for certain self-adjoint operators with applications to differential equations, J. Differential Equations, 181(2), 2002, 280-339.
- Index entries for linear recurrences with constant coefficients, signature (40,-508,2304,-2880).
Programs
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GAP
List([4..20], n-> 2^(n-7)*(20*3^n - 7*6^n + 10^n - 28)/315); # G. C. Greubel, Mar 16 2019
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Magma
[2^(n-7)*(20*3^n - 7*6^n + 10^n - 28)/315: n in [4..20]]; // G. C. Greubel, Mar 16 2019
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Mathematica
Flatten[ Table[ Sum[(-1)^{r + 4}(2r + 1)(r^2 + r)^n/((r + 5)!(4 - r)!), {r, 1, 4}], {n, 4, 20}]] LinearRecurrence[{40, -508, 2304, -2880}, {1, 40, 1092, 25664}, 20] (* G. C. Greubel, Mar 16 2019 *) -
PARI
{a(n) = 2^(n-7)*(20*3^n - 7*6^n + 10^n - 28)/315}; \\ G. C. Greubel, Mar 16 2019 -
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
Formula
From Wolfdieter Lang, Nov 07 2003: (Start)
a(n+4) = A071951(n+4, 4) = (-7*2^n + 405*6^n - 2268*12^n + 2500*20^n)/630, n >= 0.
G.f.: x^4/((1-2*1*x)*(1-3*2*x)*(1-4*3*x)*(1-5*4*x)). (End)
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
From G. C. Greubel, Mar 16 2019: (Start)
a(n) = 2^(n-7)*(20*3^n - 7*6^n + 10^n - 28)/315.
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)
Extensions
More terms from Robert G. Wilson v, Jun 19 2002
Definition corrected by Georg Fischer, Jul 07 2025