A002429 Numerators of double sums of reciprocals. (Formerly M4956 N2124)

1, 1, 14, 818, 141, 13063, 16774564, 1057052, 4651811, 778001383, 1947352646, 1073136102266, 72379420806883, 112229882767, 120372921248744, 13224581478608216, 2077531074698521033, 517938126297258811, 13785854249175914469406, 343586489824688536178, 1958290344469311726833

Offset

0,3

Comments

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

References

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

G. C. Greubel, Table of n, a(n) for n = 0..770

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.

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

Formula

a(n) = 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, 25}] (* Jean-François Alcover, Nov 04 2013 *)
a[n_]:= Numerator[3*Sum[2^(k-2)*Binomial[2*(n+1), k-1]*StirlingS1[k, 3]/k!, {k, 3, 2*n+3}]]; Table[a[n], {n, 0, 25}] (* G. C. Greubel, Jul 03 2019 *)

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
(Magma) [Numerator(3*(&+[2^(k-2)*Binomial(2*(n+1), k-1)* StirlingFirst(k, 3)/Factorial(k): k in [3..2*n+3]]) ): n in [0..25]]; // G. C. Greubel, Jul 03 2019
(Sage) [numerator( 3*sum((-1)^(k-1)*2^(k-2)*binomial(2*(n+1), k-1)* stirling_number1(k, 3)/factorial(k) for k in (3..2*n+3)) ) for n in (0..25)] # G. C. Greubel, Jul 03 2019
(GAP) List([0..25], n-> NumeratorRat( 3*Sum([3..2*n+3], k-> (-1)^(k-1)*2^(k-2)* Binomial(2*(n+1), k-1)*Stirling1(k, 3)/Factorial(k)) )) # G. C. Greubel, Jul 03 2019

Crossrefs

Cf. A008309, A049218.

Sequence in context: A050983 A360612 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