OFFSET
4,2
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).
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)
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 *)
PROG
(PARI) {a(n) = 2^(n-7)*(20*3^n - 7*6^n + 10^n - 28)/315}; \\ G. C. Greubel, Mar 16 2019
(Magma) [2^(n-7)*(20*3^n - 7*6^n + 10^n - 28)/315: n in [4..20]]; // G. C. Greubel, Mar 16 2019
(SageMath) [2^(n-7)*(20*3^n - 7*6^n + 10^n - 28)/315 for n in (4..20)] # G. C. Greubel, Mar 16 2019
(GAP) List([4..20], n-> 2^(n-7)*(20*3^n - 7*6^n + 10^n - 28)/315); # G. C. Greubel, Mar 16 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jun 16 2002
EXTENSIONS
More terms from Robert G. Wilson v, Jun 19 2002
Definition corrected by Georg Fischer, Jul 07 2025
STATUS
approved