A049034
Scaled sums of odd reciprocals.
Original entry on oeis.org
1, 8, 184, 8448, 648576, 74972160, 12174658560, 2643856588800, 740051782041600, 259500083163955200, 111422936937037824000, 57504006817918746624000, 35122852492484487413760000
Offset: 0
Joe Keane (jgk(AT)jgk.org)
(arctanh x)^2 = x^2 + 2/3*x^4 + 23/45*x^6 + 44/105*x^8 + ...
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Module[{nn=25,c},c=Range[1,nn,2];Times@@@Thread[{Accumulate[1/c],c!}]](* Harvey P. Dale, Nov 20 2013 *)
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{a(n)=if(n<0, 0, n=2*n+1; n!*sum(k=1, n, (k%2)/k))} /* Michael Somos, Sep 19 2006 */
A109792
Expansion of e.g.f. log(1+x)/(1-x)^2.
Original entry on oeis.org
1, 3, 14, 70, 444, 3108, 25584, 230256, 2342880, 25771680, 312888960, 4067556480, 57424792320, 861371884800, 13869128448000, 235775183616000, 4264876094976000, 81032645804544000, 1627055289796608000, 34168161085728768000, 754132445894209536000, 17345046255566819328000
Offset: 1
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CoefficientList[Series[Log[1+x]/(1-x)^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 27 2013 *)
a[n_] := n! ((-1)^n (n + 1) LerchPhi[-1, 1, n + 2] + Log[2] (n + 1) + ((-1)^(n + 1) - 1) /2); Table[Simplify[a[n]], {n, 1, 22}] (* Peter Luschny, Jun 22 2022 *)
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for(n=1,25, print1(n!*sum(k=1,n, sum(i=1, k, (-1)^(i+1)/i)), ", ")) \\ G. C. Greubel, Jan 21 2017
A330016
a(n) = Sum_{k=1..n} (-1)^(n - k) * H(k) * k!, where H(k) is the k-th harmonic number.
Original entry on oeis.org
0, 1, 2, 9, 41, 233, 1531, 11537, 98047, 928529, 9700111, 110843729, 1375599151, 18427159889, 265038487471, 4074124514129, 66660157879471, 1156745432699729, 21220242625821871, 410344904191816529, 8342603132569783471, 177902207647600456529, 3970574571687854263471
Offset: 0
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Table[Sum[(-1)^(n - k) HarmonicNumber[k] k!, {k, 1, n}], {n, 0, 22}]
Showing 1-3 of 3 results.
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