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 A140351 Numerator of the coefficient [x^1] of the Bernoulli twin number polynomial C(n,x). 7
 1, 0, -1, -1, -1, 1, 1, -1, -3, 3, 5, -5, -691, 691, 35, -35, -3617, 3617, 43867, -43867, -1222277, 1222277, 854513, -854513, -1181820455, 1181820455, 76977927, -76977927, -23749461029, 23749461029, 8615841276005, -8615841276005, -84802531453387, 84802531453387 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,9 COMMENTS The Bernoulli twin number polynomials C(n,x) are defined in A129378. I call the full fraction [x^1]C(n,x) the secondary Bernoulli twin numbers. LINKS FORMULA a(n) = numerator(Sum_{i=0..n} binomial(n,i)*(i+1)*bernoulli(i)). - Vladimir Kruchinin, Oct 05 2016 EXAMPLE The coefficients [x^m]C(n,x) are a table of fractions: 1 ; -1/2, 1; -1/3, 0, 1; -1/6, -1/2, 1/2, 1; -1/30,-1/2, -1/2, 1, 1; 1/30, -1/6, -1,-1/3, 3/2, 1; 1/42, 1/6, -1/2, -5/3, 0, 2, 1; -1/42, 1/6, 1/2, -7/6, -5/2, 1/2, 5/2, 1; -1/30, -1/6, 2/3, 7/6, -7/3, -7/2, 7/6, 3, 1; 1/30, -3/10, -2/3, 2, 7/3, -21/5, -14/3, 2, 7/2, 1; 5/66, 3/10, -3/2, -2, 5, 21/5, -7, -6, 3, 4, 1; ... This sequence here contains the numerators of the second column. MAPLE 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: A140351 := proc(n) coeff(C(n, x), x, 1) ; numer(%) ; end proc: seq(A140351(n), n=1..80) ; # R. J. Mathar, Nov 22 2009 MATHEMATICA 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 *) Table[Sum[Binomial[n, k]*(k+1)*BernoulliB[k], {k, 0, n}], {n, 0, 30}] // Numerator (* Vaclav Kotesovec, Oct 05 2016 *) PROG (Maxima) makelist(num(sum((binomial(n, i)*(i+1)*bern(i)), i, 0, n)), n, 0, 20); /* Vladimir Kruchinin, Oct 05 2016 */ (PARI) a(n) = numerator(sum(i=0, n, binomial(n, i)*(i+1)*bernfrac(i))); \\ Michel Marcus, Oct 05 2016 CROSSREFS Cf. A129378, A129826. Sequence in context: A087745 A087756 A243305 * A128444 A204250 A131948 Adjacent sequences:  A140348 A140349 A140350 * A140352 A140353 A140354 KEYWORD frac,sign AUTHOR Paul Curtz, May 30 2008, Jun 23 2008 EXTENSIONS Edited and extended by R. J. Mathar, Nov 22 2009 STATUS approved

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