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A071952 Diagonal T(n,3) of triangle in A071951. 6
1, 40, 1092, 25664, 561104, 11807616, 243248704, 4950550528, 100040447232, 2013177300992, 40412056994816, 810023815790592, 16221871691714560, 324694197936160768, 6496965245491888128, 129976281056339296256 (list; graph; refs; listen; history; text; internal format)
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

(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

(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

Cf. A000079, A016129, A015309, A089278, A089500.

Cf. A071951, A071952.

Sequence in context: A062143 A284838 A124100 * A331906 A144914 A004385

Adjacent sequences:  A071949 A071950 A071951 * A071953 A071954 A071955

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jun 16 2002

EXTENSIONS

More terms from Robert G. Wilson v, Jun 19 2002

STATUS

approved

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Last modified February 28 01:54 EST 2020. Contains 332319 sequences. (Running on oeis4.)