A002429 Numerators of double sums of reciprocals.
1, 1, 14, 818, 141, 13063, 16774564, 1057052, 4651811, 778001383, 1947352646, 1073136102266, 72379420806883, 112229882767, 120372921248744, 13224581478608216, 2077531074698521033, 517938126297258811, 13785854249175914469406, 343586489824688536178, 1958290344469311726833
Offset: 0
Keywords
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.
Programs
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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
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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
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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
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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 *) -
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
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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
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
Extensions
More terms from Ruperto Corso, Dec 09 2011
Comments