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A137777
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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.
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2
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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; internal format)
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OFFSET
| 0,1
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COMMENTS
| Row sums are: {2, 2, 0, -8, 0, 240, 0, -24192, 0, 6048000, 0, ...}.
Contribution from Peter Luschny (peter(AT)luschny.de), 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)
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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 (mathar(AT)strw.leidenuniv.nl), Jun 03 2009]
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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}
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MAPLE
| seq(seq(coeff(bernoulli(k, x)*2*(k+1)!, x, i), i=0..k), k=1..10); [From Peter Luschny (peter(AT)luschny.de), Apr 23 2009]
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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
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CROSSREFS
| Sequence in context: A067228 A010026 A059427 * A126984 A159749 A102416
Adjacent sequences: A137774 A137775 A137776 * A137778 A137779 A137780
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KEYWORD
| tabl,sign
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AUTHOR
| Roger L. Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Apr 28 2008
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EXTENSIONS
| Edited by N. J. A. Sloane, Jan 03 2010, incorprating comments from Peter Luschny and Peter Pein
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