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A137777 Triangular sequence of coefficients from the expansion of the derivative of the Bernoulli polynomial function: p(x,t) = t*exp(x*t)/(exp(t)-1); q(x,t) = p'(x,t) = dp(x,t)/dt.

%I A137777 #16 Mar 17 2018 05:39:33
%S A137777 2,-2,4,2,-12,12,0,24,-72,48,-8,0,240,-480,240,0,-240,0,2400,-3600,
%T A137777 1440,240,0,-5040,0,25200,-30240,10080,0,13440,0,-94080,0,282240,
%U A137777 -282240,80640,-24192,0,483840,0,-1693440,0,3386880,-2903040,725760,0,-2177280,0,14515200,0,-30481920,0,43545600,-32659200
%N A137777 Triangular sequence of coefficients from the expansion of the derivative of the Bernoulli polynomial function: p(x,t) = t*exp(x*t)/(exp(t)-1); q(x,t) = p'(x,t) = dp(x,t)/dt.
%C A137777 Row sums are {2, 2, 0, -8, 0, 240, 0, -24192, 0, 6048000, 0, ...}.
%C A137777 From _Peter Luschny_, Apr 23 2009: (Start)
%C A137777 The sequence can also be computed as the coefficients of the Bernoulli polynomials B_n(x) times 2(n+1)! for n >= 1. As Peter Pein observed the Mathematica code then reduces to
%C A137777 Table[CoefficientList[2 (n+1)! BernoulliB[n,x],x],{n,1,10}] // Flatten
%C A137777 Note that this formula is also well defined in the case n = 0 and has the value 2. (End)
%F A137777 p(x,t) = t*exp(x*t)/(exp(t)-1); q(x,t) = p'(x,t) = dp(x,t)/dt = Sum_{n>=0} Q(x,n)*t^n/n!; out_n,m=2*(n + 2)!*n!*Coefficients(Q(x,n).
%F A137777 A137777(n,0) = 2*A129814(n) for n >= 0.
%F A137777 A137777(n,n) = 2*(n+1)! for n >= 0.
%F A137777 Conjecture on row sums: Sum_{k=0..n+1} T(n,k) = 2*A129825(n+2). - _R. J. Mathar_, Jun 03 2009
%e A137777 {2},
%e A137777 {-2, 4},
%e A137777 {2, -12, 12},
%e A137777 {0,24, -72, 48},
%e A137777 {-8, 0, 240, -480, 240},
%e A137777 {0, -240, 0, 2400, -3600, 1440},
%e A137777 {240, 0, -5040, 0, 25200, -30240, 10080},
%e A137777 {0, 13440, 0, -94080, 0, 282240, -282240, 80640},
%e A137777 {-24192, 0, 483840, 0, -1693440, 0, 3386880, -2903040, 725760},
%e A137777 {0, -2177280, 0, 14515200, 0, -30481920, 0, 43545600, -32659200, 7257600},
%e A137777 {6048000, 0, -119750400, 0, 399168000, 0, -558835200, 0, 598752000, -399168000, 79833600},
%e A137777 {0, 798336000, 0, -5269017600, 0, 10538035200, 0, -10538035200, 0, 8781696000, -5269017600, 958003200}
%p A137777 seq(seq(coeff(bernoulli(k,x)*2*(k+1)!,x,i),i=0..k),k=1..10); # _Peter Luschny_, Apr 23 2009
%t A137777 Clear[p, b, a]; p[t_] = D[t^2*Exp[x*t]/(Exp[t]-1),{t,1}];
%t A137777 a = Table[CoefficientList[2*n!^2*SeriesCoefficient
%t A137777 [Series[p[t],{t,0,30}],n],x],{n,0,10}]; Flatten[a]
%t A137777 Table[CoefficientList[2 BernoulliB[k,x] Gamma[2+k],x],{k,0,10}]//Flatten
%K A137777 tabl,sign
%O A137777 0,1
%A A137777 _Roger L. Bagula_ and _Gary W. Adamson__, Apr 28 2008
%E A137777 Edited by _N. J. A. Sloane_, Jan 03 2010, incorporating comments from _Peter Luschny_ and Peter Pein