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%I M4956 N2124 #36 Sep 08 2022 08:44:30
%S 1,1,14,818,141,13063,16774564,1057052,4651811,778001383,1947352646,
%T 1073136102266,72379420806883,112229882767,120372921248744,
%U 13224581478608216,2077531074698521033,517938126297258811,13785854249175914469406,343586489824688536178,1958290344469311726833
%N Numerators of double sums of reciprocals.
%C Also, numerators of coefficients of expansion of arctan(x)^3. - _Ruperto Corso_, Dec 09 2011
%D 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.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H G. C. Greubel, <a href="/A002429/b002429.txt">Table of n, a(n) for n = 0..770</a>
%H Mohammad K. Azarian, <a href="https://www.jstor.org/stable/2686918">A Double Sum, Problem 440</a>, College Mathematics Journal, Vol. 21, No. 5, Nov. 1990, p. 424. <a href="https://www.jstor.org/stable/2686609">Solution</a> published in Vol. 22. No. 5, Nov. 1991, pp. 448-449.
%H H. A. Rothe, in C. F. Hindenburg, editor, <a href="http://www.archive.org/stream/sammlungcombina03hindgoog#page/n373/mode/1up">Sammlung Combinatorisch-Analytischer Abhandlungen</a>, Vol. 2, Chap. XI. Fleischer, Leipzig, 1800, p. 316.
%F 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
%p 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;
%p 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;
%p A002429 := proc(n)
%p numer(p3x(n)) ;
%p end proc:
%p seq(A002429(n),n=1..25) ; # _Ruperto Corso_, Dec 09 2011
%t 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 *)
%t 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 *)
%o (PARI) stirling1(n, k)=if(n<1, 0, n!*polcoeff(binomial(x, n), k))
%o 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
%o (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
%o (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
%o (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
%Y Cf. A008309, A049218.
%K nonn
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _Ruperto Corso_, Dec 09 2011
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