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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. 2

%I

%S 2,-2,4,2,-12,12,0,24,-72,48,-8,0,240,-480,240,0,-240,0,2400,-3600,

%T 1440,240,0,-5040,0,25200,-30240,10080,0,13440,0,-94080,0,282240,

%U -282240,80640,-24192,0,483840,0,-1693440,0,3386880,-2903040,725760,0,-2177280,0,14515200,0,-30481920,0,43545600,-32659200

%N 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 Row sums are {2, 2, 0, -8, 0, 240, 0, -24192, 0, 6048000, 0, ...}.

%C From _Peter Luschny_, Apr 23 2009: (Start)

%C 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 Table[CoefficientList[2 (n+1)! BernoulliB[n,x],x],{n,1,10}] // Flatten

%C Note that this formula is also well defined in the case n = 0 and has the value 2. (End)

%F 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(n,0) = 2*A129814(n) for n >= 0.

%F A137777(n,n) = 2*(n+1)! for n >= 0.

%F Conjecture on row sums: Sum_{k=0..n+1} T(n,k) = 2*A129825(n+2). - _R. J. Mathar_, Jun 03 2009

%e {2},

%e {-2, 4},

%e {2, -12, 12},

%e {0,24, -72, 48},

%e {-8, 0, 240, -480, 240},

%e {0, -240, 0, 2400, -3600, 1440},

%e {240, 0, -5040, 0, 25200, -30240, 10080},

%e {0, 13440, 0, -94080, 0, 282240, -282240, 80640},

%e {-24192, 0, 483840, 0, -1693440, 0, 3386880, -2903040, 725760},

%e {0, -2177280, 0, 14515200, 0, -30481920, 0, 43545600, -32659200, 7257600},

%e {6048000, 0, -119750400, 0, 399168000, 0, -558835200, 0, 598752000, -399168000, 79833600},

%e {0, 798336000, 0, -5269017600, 0, 10538035200, 0, -10538035200, 0, 8781696000, -5269017600, 958003200}

%p seq(seq(coeff(bernoulli(k,x)*2*(k+1)!,x,i),i=0..k),k=1..10); # _Peter Luschny_, Apr 23 2009

%t Clear[p, b, a]; p[t_] = D[t^2*Exp[x*t]/(Exp[t]-1),{t,1}];

%t a = Table[CoefficientList[2*n!^2*SeriesCoefficient

%t [Series[p[t],{t,0,30}],n],x],{n,0,10}]; Flatten[a]

%t Table[CoefficientList[2 BernoulliB[k,x] Gamma[2+k],x],{k,0,10}]//Flatten

%K tabl,sign

%O 0,1

%A _Roger L. Bagula_ and _Gary W. Adamson__, Apr 28 2008

%E Edited by _N. J. A. Sloane_, Jan 03 2010, incorporating comments from _Peter Luschny_ and Peter Pein

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Last modified June 15 12:29 EDT 2021. Contains 345048 sequences. (Running on oeis4.)