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A260336 Numerators of Glaisher's J-numbers J_n.

%I A260336 #19 Apr 15 2023 05:03:39
%S A260336 10,34,910,415826,3786350,455594594,226816276970,16546152735874,
%T A260336 4616987879606830,4799607558341375462,674014218452089817870,
%U A260336 339274220304210587466434,5429636257086663655134162970,138634566648793083166951423714
%N A260336 Numerators of Glaisher's J-numbers J_n.
%H A260336 J. W. L. Glaisher, <a href="https://doi.org/10.1112/plms/s1-31.1.216">On a set of coefficients analogous to the Eulerian numbers</a>, Proc. London Math. Soc., 31 (1899), 216-235.
%H A260336 <a href="/index/Ge#Glaisher">Index entries for sequences related to Glaisher's numbers</a>
%e A260336 10/3, 34, 910, 415826/9, 3786350, 455594594, 226816276970/3, 16546152735874, 4616987879606830, ...
%p A260336 In := proc(n)
%p A260336     1/(exp(x)+exp(-x)+1) ;
%p A260336     coeftayl(%,x=0,2*n) ;
%p A260336     %*(2*n)!*(-1)^n*3/2 ;
%p A260336 end proc:
%p A260336 Jn := proc(n)
%p A260336     (2^(2*n+1)+2)*In(n) ;
%p A260336 end proc:
%p A260336 A260336 := proc(n)
%p A260336     numer(Jn(n)) ;
%p A260336 end proc: # _R. J. Mathar_, Aug 02 2015
%t A260336 GI[n_] := SeriesCoefficient[1/(Exp[x]+Exp[-x]+1), {x, 0, 2n}]*(2n)!*(-1)^n*3/2;
%t A260336 GJ[n_] := (2^(2n+1)+2)*GI[n];
%t A260336 a[n_] := Numerator[GJ[n]];
%t A260336 Table[a[n], {n, 1, 14}] (* _Jean-François Alcover_, Apr 15 2023, after _R. J. Mathar_ *)
%Y A260336 Cf. A047789 (denominators).
%K A260336 nonn,frac
%O A260336 1,1
%A A260336 _N. J. A. Sloane_, Jul 29 2015