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A145609
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Numerator of the polynomial A_l(x) = sum_{d=1..l-1} x^(l-d)/d for index l=2n+1 evaluated at x=1.
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41
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3, 25, 49, 761, 7381, 86021, 1171733, 2436559, 14274301, 55835135, 19093197, 1347822955, 34395742267, 315404588903, 9304682830147, 586061125622639, 54062195834749, 54801925434709, 2053580969474233, 2078178381193813
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| For denominators see A145610.
The polynomials A_{2n+1}(x) = sum_{d=1..2n} x^(2n+1-d)/d for small n look as follows:
n=1, index = 3: A_3(x) = x/2 + x^2.
n=2, index = 5: A_5(x) = x/4 + x^2/3 + x^3/2 + x^4.
n=3, index = 7: A_7(x) = x/6 + x^2/5 + x^3/4 + x^4/3 + x^5/2 + x^6.
n=4, index = 9: A_9(x) = x/8 + x^2/7 + x^3/6 + x^4/5 + x^5/4 + x^6/3 + x^7/2 + x^8.
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FORMULA
| (1/(2n+1))*2F1(1,2n+1;2n+2;1/m) = Sum_{x=0..infinity} m^(-x)/(x+2n+1) =
m^(2n)arctanh((2m-1)/(2m^2-2m+1))-A_{2n+1}(m) = m^(2n)log(m/(m-1))-A_{2n+1}(m) [Artur Jasinski, Oct 14 2008]
It appears that A145609(n)/A145610(n)=H(2*n+2), the harmonic number of order 2*n+2
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MAPLE
| A := proc(l, x) add(x^(l-d)/d, d=1..l-1) ; end: A145609 := proc(n) numer( A(2*n+1, 1)) ; end: seq(A145609(n), n=1..20) ; # R. J. Mathar, Aug 21 2009
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MATHEMATICA
| m = 1; aa = {}; Do[k = 0; Do[k = k + m^(2 r + 1 - d)/d, {d, 1, 2 r}]; AppendTo[aa, Numerator[k]], {r, 1, 25}]; aa (*Artur Jasinski*)
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CROSSREFS
| Cf. A145609 - A145640.
Sequence in context: A129599 A042899 A051280 * A120285 A041897 A006222
Adjacent sequences: A145606 A145607 A145608 * A145610 A145611 A145612
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KEYWORD
| frac,nonn
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AUTHOR
| Artur Jasinski (grafix(AT)csl.pl), Oct 14 2008
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EXTENSIONS
| Edited, parentheses in front of Gauss. Hypg. Fct. added by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 21 2009
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