A140351 Numerator of the coefficient [x^1] of the Bernoulli twin number polynomial C(n,x).
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
Offset: 1
Examples
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.
Programs
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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
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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 *) -
Maxima
makelist(num(sum((binomial(n,i)*(i+1)*bern(i)),i,0,n)),n,0,20); /* Vladimir Kruchinin, Oct 05 2016 */
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PARI
a(n) = numerator(sum(i=0, n, binomial(n,i)*(i+1)*bernfrac(i))); \\ Michel Marcus, Oct 05 2016
Formula
a(n) = numerator(Sum_{i=0..n} binomial(n,i)*(i+1)*bernoulli(i)). - Vladimir Kruchinin, Oct 05 2016
Extensions
Edited and extended by R. J. Mathar, Nov 22 2009
Comments