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A227985 Numerators of the fractional triangle T(n,k) = binomial(n-1,k)*B_k for 0 <= k < n.

%I A227985 #54 Sep 08 2022 08:46:05
%S A227985 1,0,-1,0,-1,1,0,-1,1,-1,0,-1,1,-1,0,0,-1,1,-5,0,1,0,-1,1,-1,0,1,0,0,
%T A227985 -1,1,-7,0,7,0,-1,0,-1,1,-2,0,7,0,-2,0,0,-1,1,-3,0,7,0,-1,0,3,0,-1,1,
%U A227985 -5,0,1,0,-1,0,1,0,0,-1,1,-11,0,11,0,-11,0,11,0,-5,0,-1,1,-1,0,11,0,-22,0,33,0,-5,0
%N A227985 Numerators of the fractional triangle T(n,k) = binomial(n-1,k)*B_k for 0 <= k < n.
%C A227985 The n-th row's sum equals the n-th Bernoulli number (with B_1 = -1/2).
%C A227985 Starting from B_0 = 1, the successive B n comes from the equations written with the triangle A074909
%C A227985 1*B_0 +2*B_1 = 0                   -->  B_1 = 0 -1/2
%C A227985 1*B_0 +3*B_1 +3*B_2 = 0            -->  B_2 = 0 -1/3 +1/2
%C A227985 1*B_0 +4*B_1 +6*B_2 +4*B_3 = 0     -->  B_3 = 0 -1/4 +1/2 -1/4,
%C A227985 from the terms at the left-hand side. See A159688.
%C A227985 Main diagonal: 1, -1/2, 1/2, -1/4, 0, 1/12, 0, -1/12, 0, 3/20, 0, -5/12, 0, 691/420,... . After the initial 1, the numerators are given by -A050925.
%e A227985 Triangle begins:
%e A227985 1,
%e A227985 0, -1,
%e A227985 0, -1, 1,
%e A227985 0, -1, 1, -1,
%e A227985 0, -1, 1, -1, 0,
%e A227985 0, -1, 1, -5, 0, 1,
%e A227985 0  -1, 1, -1, 0, 1,  0,
%e A227985 0, -1, 1, -7, 0, 7,  0, -1,
%e A227985 0, -1, 1, -2, 0, 7,  0, -2, 0, etc.
%t A227985 b[0] = 1; b[1] = -1/2; row[0] = {1}; row[1] = {0, -1/2}; row[n_] := Join[{0}, List @@ (-Sum[Binomial[n+1, k]*B[k], {k, 0, n-1}]/(n+1) // Expand) /. B -> b]; b[n_] := Total[row[n]]; Table[row[n] // Numerator, {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Aug 12 2013 *)
%o A227985 (PARI) t(n, k) = if (n==1, 1, if (k== -1, 0, -bernfrac(k)*binomial(n, k)/n));
%o A227985 tabl(nn) = {for (n = 1, nn, for (k = -1, n-2, print1(t(n, k), ", ");); print(););} \\ _Michel Marcus_, Sep 07 2013
%o A227985 (Magma) [1] cat [Numerator(-Binomial(n,k)*Bernoulli(k)/n): k in [-1..n-2], n in [2..15]]; // _Bruno Berselli_, Sep 09 2013
%Y A227985 Cf. A027641/A027642, A074909.
%K A227985 sign,frac,tabl
%O A227985 0,19
%A A227985 _Paul Curtz_, Aug 02 2013
%E A227985 More terms from _Jean-François Alcover_, Aug 12 2013