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A002429 Numerators of double sums of reciprocals.
(Formerly M4956 N2124)
2
1, 1, 14, 818, 141, 13063, 16774564, 1057052, 4651811, 778001383, 1947352646, 1073136102266, 72379420806883, 112229882767, 120372921248744, 13224581478608216, 2077531074698521033, 517938126297258811, 13785854249175914469406, 343586489824688536178, 1958290344469311726833 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also, numerators of coefficients of expansion of arctan(x)^3. - Ruperto Corso, Dec 09 2011

REFERENCES

Mohammad K. Azarian, A Double Sum, Problem 440, College Mathematics Journal, Vol. 21, No. 5, Nov. 1990, p. 424.  Solution published in Vol. 22. No. 5, Nov. 1991, pp. 448-449.

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 117.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=0..20.

H. A. Rothe, in C. F. Hindenburg, editor, Sammlung Combinatorisch-Analytischer Abhandlungen, Vol. 2, Chap. XI. Fleischer, Leipzig, 1800, p. 316.

FORMULA

a(n) is the numerator of 3*sum_{i=3..2*n+3} 2^(i-2)*binomial(2*(n+1),i-1)*stirling1(i,3)/ i!. - Ruperto Corso, Dec 09 2011

MAPLE

p2x:=proc(n) option remember: if(n=1) then RETURN(1) else RETURN(((n-1)*p2x(n-1)+1/(2*n-1))/n) fi: end proc;

p3x:=proc(n) option remember: if(n=1) then RETURN(1) else RETURN(((2*n-1)*p3x(n-1)+3*p2x(n))/(2*n+1)) fi: end proc;

A002429 := proc(n)

    numer(p3x(n)) ;

end proc:

seq(A002429(n), n=1..25) ; # Ruperto Corso, Dec 09 2011

MATHEMATICA

a[n_] := (-1)^n*SeriesCoefficient[ ArcTan[x]^3, {x, 0, 2*n+3}] // Numerator; Table[a[n], {n, 0, 20}] (* Jean-Fran├žois Alcover, Nov 04 2013 *)

PROG

(PARI) stirling1(n, k)=if(n<1, 0, n!*polcoeff(binomial(x, n), k))

for(n=0, 25, print1(numerator(3/4*sum(i=3, 2*n+3, 2^i*binomial(2*(n+1), i-1)*stirling1(i, 3)/ i!))", ")) /* Ruperto Corso, Dec 09 2011 */

CROSSREFS

Cf. A008309, A049218.

Sequence in context: A042519 A050983 A183576 * A064345 A269335 A159871

Adjacent sequences:  A002426 A002427 A002428 * A002430 A002431 A002432

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Ruperto Corso, Dec 09 2011

STATUS

approved

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Last modified October 20 01:39 EDT 2018. Contains 316378 sequences. (Running on oeis4.)