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A213616 Triangle read by rows, coefficients of the Bernoulli nabla polynomials BN_{n}(x) times A144845(n) in descending order of powers.

%I A213616 #13 Sep 27 2013 06:11:58
%S A213616 1,2,-3,6,-18,13,2,-9,13,-6,30,-180,390,-360,119,6,-45,130,-180,119,
%T A213616 -30,42,-378,1365,-2520,2499,-1260,253,6,-63,273,-630,833,-630,253,
%U A213616 -42,30,-360,1820,-5040,8330,-8400,5060,-1680,239,10,-135,780,-2520,4998,-6300
%N A213616 Triangle read by rows, coefficients of the Bernoulli nabla polynomials BN_{n}(x) times A144845(n) in descending order of powers.
%C A213616 The polynomials BN_{n}(x) have the e.g.f. t*exp(t*(x-1))/(exp(t)-1). The adjunct 'nabla' in the name refers to the backward difference operation.
%C A213616 BN_{n}(1) are the Bernoulli numbers.
%C A213616 In the difference table of the Bernoulli polynomials the polynomials BN_{n}(x) appear as the top row (see the link).
%H A213616 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/ComputationAndAsymptoticsOfBernoulliNumbers">The Computation and Asymptotics of the Bernoulli numbers.</a>
%F A213616 T(n,k) = A144845(n)*[x^(n-k)]BN{n}(x).
%e A213616 bn(0,x) =  1,
%e A213616 bn(1,x) =  2*x   -   3,
%e A213616 bn(2,x) =  6*x^2 -  18*x   +  13,
%e A213616 bn(3,x) =  2*x^3 -   9*x^2 +  13*x   -   6,
%e A213616 bn(4,x) = 30*x^4 - 180*x^3 + 390*x^2 - 360*x   + 119,
%e A213616 bn(5,x) =  6*x^5 -  45*x^4 + 130*x^3 - 180*x^2 + 119*x - 30.
%p A213616 seq(seq(coeff(denom(bernoulli(i, x))*bernoulli(i, x - 1), x, i - j), j=0..i), i=0..12);
%t A213616 Table[If[i == 0, 1, 1/First[ FactorTerms[ BernoulliB[i, x]]]]*Table[ Coefficient[ Denominator[ BernoulliB[i, x]]*BernoulliB[i, x-1], x, i-j], {j, 0, i}], {i, 0, 12}] // Flatten (* _Jean-François Alcover_, Sep 27 2013, after Maple *)
%Y A213616 A053383, A144845, A213615.
%K A213616 sign,tabl
%O A213616 0,2
%A A213616 _Peter Luschny_, Jun 16 2012