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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
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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

Contribution 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 Pain 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)

LINKS

Table of n, a(n) for n=0..53.

FORMULA

p(x,t)=t*Exp(x*t)/(Exp(t)-1); q(x,t)=p'(x,t)=dp(x,t)/dt=Sum(Q(x,n)*t^n/n!,{n,0,Infinity}); 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). [From R. J. Mathar, Jun 03 2009]

EXAMPLE

{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}

MAPLE

seq(seq(coeff(bernoulli(k, x)*2*(k+1)!, x, i), i=0..k), k=1..10); [From 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

CROSSREFS

Sequence in context: A227450 A010026 A059427 * A126984 A159749 A227293

Adjacent sequences:  A137774 A137775 A137776 * A137778 A137779 A137780

KEYWORD

tabl,sign

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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Last modified November 22 06:11 EST 2017. Contains 295076 sequences.