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A122057
a(n) = (n+1)! * (H(n+1) - H(2)), where H(n) are the harmonic numbers.
1
0, 2, 14, 94, 684, 5508, 49104, 482256, 5185440, 60668640, 767940480, 10462227840, 152698210560, 2377651449600, 39350097561600, 689874448435200, 12773427499929600, 249097496204390400, 5103595024496640000, 109608397522606080000
OFFSET
1,2
COMMENTS
Former title (corrected): A Legendre-based recurrence sequence. Let b(n) = ((4*n+2)*x -(2*n+1) )/(n+1)*b(n-1) - (n/(n+1))*b(n-2), where x=1, then a(n) = (n+1)!*b(n)/6. - G. C. Greubel, Oct 03 2019
REFERENCES
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964, 9th Printing (1970), pp. 782
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
FORMULA
Let b(n) = ((-2*n-1) +(4*n+2)*x)/(n+1)*b(n-1) - (n/(n+1))*b(n-2) with x=1, then a(n) = b(n)*(n+1)!/6.
a(n) = (n+1)! * Sum_{k=3..n+1} 1/k. - Gary Detlefs, Jul 15 2010
MAPLE
a:=n-> (n+1)!*add(1/k, k=3..n+1): seq(a(n), n=1..30); # Gary Detlefs, Jul 15 2010
MATHEMATICA
x=1; b[1]:=0; b[2]:=2; b[n_]:= b[n]= ((-2*n-1) +(4*n+2)*x)/(n+1)*b[n-1] - (n/(n+1))*b[n-2]; Table[b[n]*(n+1)!/6, {n, 30}]
Table[(n+1)!*(HarmonicNumber[n+1] - 3/2), {n, 30}] (* G. C. Greubel, Oct 03 2019 *)
PROG
(PARI) vector(30, n, (n+1)!*(sum(k=1, n+1, 1/k) - 3/2) ) \\ G. C. Greubel, Oct 03 2019
(Magma) [Factorial(n+1)*(HarmonicNumber(n+1) - 3/2): n in [1..30]]; // G. C. Greubel, Oct 03 2019
(Sage) [factorial(n+1)*(harmonic_number(n+1) - 3/2) for n in (1..30)] # G. C. Greubel, Oct 03 2019
(GAP) List([1..30], n-> Factorial(n+1)*(Sum([1..n+1], k-> 1/k) - 3/2) ); # G. C. Greubel, Oct 03 2019
CROSSREFS
Sequence in context: A033169 A090410 A066052 * A164891 A141146 A267913
KEYWORD
nonn
AUTHOR
Roger L. Bagula, Sep 14 2006
EXTENSIONS
If all terms are really negative, sequence should probably be negated. - N. J. A. Sloane, Oct 01 2006
Negated terms and edited by G. C. Greubel, Oct 03 2019
STATUS
approved