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, -2, 4, 2, -12, 12, 0, 24, -72, 48, -8, 0, 240, -480, 240, 0, -240, 0, 2400, -3600, 1440, 240, 0, -5040, 0, 25200, -30240, 10080, 0, 13440, 0, -94080, 0, 282240, -282240, 80640, -24192, 0, 483840, 0, -1693440, 0, 3386880, -2903040, 725760, 0, -2177280, 0, 14515200, 0, -30481920, 0, 43545600, -32659200

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson_, Apr 28 2008

Keywords

• tabl
• sign

Comments

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

From Peter Luschny, Apr 23 2009: (Start)

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

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

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

Examples

			{2},
{-2, 4},
{2, -12, 12},
{0,24, -72, 48},
{-8, 0, 240, -480, 240},
{0, -240, 0, 2400, -3600, 1440},
{240, 0, -5040, 0, 25200, -30240, 10080},
{0, 13440, 0, -94080, 0, 282240, -282240, 80640},
{-24192, 0, 483840, 0, -1693440, 0, 3386880, -2903040, 725760},
{0, -2177280, 0, 14515200, 0, -30481920, 0, 43545600, -32659200, 7257600},
{6048000, 0, -119750400, 0, 399168000, 0, -558835200, 0, 598752000, -399168000, 79833600},
{0, 798336000, 0, -5269017600, 0, 10538035200, 0, -10538035200, 0, 8781696000, -5269017600, 958003200}
		

Programs

Maple

seq(seq(coeff(bernoulli(k,x)*2*(k+1)!,x,i),i=0..k),k=1..10); # Peter Luschny, Apr 23 2009

Mathematica

Clear[p, b, a]; p[t_] = D[t^2*Exp[x*t]/(Exp[t]-1),{t,1}];
a = Table[CoefficientList[2*n!^2*SeriesCoefficient
[Series[p[t],{t,0,30}],n],x],{n,0,10}]; Flatten[a]
Table[CoefficientList[2 BernoulliB[k,x] Gamma[2+k],x],{k,0,10}]//Flatten

Formula

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).

A137777(n,0) = 2*A129814(n) for n >= 0.

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

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

Extensions

Edited by N. J. A. Sloane, Jan 03 2010, incorporating comments from Peter Luschny and Peter Pein