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%I #28 Nov 29 2022 05:59:44
%S 1,0,-1,-1,-1,1,1,-1,-3,3,5,-5,-691,691,35,-35,-3617,3617,43867,
%T -43867,-1222277,1222277,854513,-854513,-1181820455,1181820455,
%U 76977927,-76977927,-23749461029,23749461029,8615841276005,-8615841276005,-84802531453387,84802531453387
%N Numerator of the coefficient [x^1] of the Bernoulli twin number polynomial C(n,x).
%C The Bernoulli twin number polynomials C(n,x) are defined in A129378.
%F a(n) = numerator(Sum_{i=0..n} binomial(n,i)*(i+1)*bernoulli(i)). - _Vladimir Kruchinin_, Oct 05 2016
%e The coefficients [x^m]C(n,x) are a table of fractions:
%e 1 ;
%e -1/2, 1;
%e -1/3, 0, 1;
%e -1/6, -1/2, 1/2, 1;
%e -1/30,-1/2, -1/2, 1, 1;
%e 1/30, -1/6, -1,-1/3, 3/2, 1;
%e 1/42, 1/6, -1/2, -5/3, 0, 2, 1;
%e -1/42, 1/6, 1/2, -7/6, -5/2, 1/2, 5/2, 1;
%e -1/30, -1/6, 2/3, 7/6, -7/3, -7/2, 7/6, 3, 1;
%e 1/30, -3/10, -2/3, 2, 7/3, -21/5, -14/3, 2, 7/2, 1;
%e 5/66, 3/10, -3/2, -2, 5, 21/5, -7, -6, 3, 4, 1; ...
%e This sequence here contains the numerators of the second column.
%p C := proc(n,x) if n = 0 then 1; else add(binomial(n-1,j-1)* bernoulli(j,x),j=1..n) ; expand(%) ; end if ; end proc:
%p A140351 := proc(n) coeff(C(n,x),x,1) ; numer(%) ; end proc: seq(A140351(n),n=1..80) ; # _R. J. Mathar_, Nov 22 2009
%t b[n_, x_] := Coefficient[ Series[ t*E^(x*t)/(E^t - 1), {t, 0, n}], t, n]*n!; c[n_, x_] := Sum[ Binomial[n-1, j-1]*b[j, x], {j, 1, n}]; t[n_, m_] := Coefficient[c[n, x], x, m]; Table[t[n, 1] // Numerator, {n, 1, 34} ] (* _Jean-François Alcover_, Mar 04 2013 *)
%t Table[Sum[Binomial[n, k]*(k+1)*BernoulliB[k], {k, 0, n}], {n, 0, 30}] // Numerator (* _Vaclav Kotesovec_, Oct 05 2016 *)
%o (Maxima) makelist(num(sum((binomial(n,i)*(i+1)*bern(i)),i,0,n)),n,0,20); /* _Vladimir Kruchinin_, Oct 05 2016 */
%o (PARI) a(n) = numerator(sum(i=0, n, binomial(n,i)*(i+1)*bernfrac(i))); \\ _Michel Marcus_, Oct 05 2016
%Y Cf. A002427, A129378, A129826.
%K frac,sign
%O 1,9
%A _Paul Curtz_, May 30 2008, Jun 23 2008
%E Edited and extended by _R. J. Mathar_, Nov 22 2009
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