A122057 a(n) = (n+1)! * (H(n+1) - H(2)), where H(n) are the harmonic numbers.

0, 2, 14, 94, 684, 5508, 49104, 482256, 5185440, 60668640, 767940480, 10462227840, 152698210560, 2377651449600, 39350097561600, 689874448435200, 12773427499929600, 249097496204390400, 5103595024496640000, 109608397522606080000, 2462475687669043200000

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

G. C. Greubel, Table of n, a(n) for n = 1..445

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

a(n) = 2*A001711(n-2) for n >= 2. - Pontus von Brömssen, Jan 04 2025

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

Cf. A001008, A002805, A001705, A001711.

Sequence in context: A033169 A090410 A066052 * A164891 A141146 A267913

Adjacent sequences: A122054 A122055 A122056 * A122058 A122059 A122060

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