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
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Ruperto Corso, Dec 09 2011
STATUS
approved