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Search: id:A137524
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| A137524 |
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Triangular sequence from coefficients of the umbral calculus expansion of a Golden -Mean Bernoulli function(A001898): p(x,t)=t*GoldenRatio^(x*t)/(GoldenRatio^t - 1), where the golden ratio replaces "e". |
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1
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| 2, -3, 6, 4, -24, 24, 0, 60, -180, 120, -24, 0, 720, -1440, 720, 0, -840, 0, 8400, -12600, 5040, 960, 0, -20160, 0, 100800, -120960, 40320, 0, 60480, 0, -423360, 0, 1270080, -1270080, 362880, -120960, 0, 2419200, 0, -8467200, 0, 16934400, -14515200, 3628800, 0, -11975040, 0, 79833600, 0, -167650560, 0
(list; table; graph; listen)
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OFFSET
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1,1
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COMMENT
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Row sums are: {2, 3, 4, 0, -24, 0, 960, 0, -120960, 0, 36288000}
These are the same as the Bernoulli numbers with the factor Log[GoldenRatio]^n: p[t_] = t*Exp[x*t]/(Exp[t] - 1);
a = Table[ CoefficientList[(n + 2)!*n!*SeriesCoefficient[Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}];
Exp[1]^Log[GoldenRatio]=GoldenRatio.
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FORMULA
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p(x,t)=t*GoldenRatio^(x*t)/(GoldenRatio^t - 1)=Sum[P(x,n)*t^n/n!,{n,0,Infinity}]; out_n,m=((n + 2)!*n!/Log[GoldenRatio]^(n-1))*Coefficients(P(x,n))
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EXAMPLE
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{2},
{-3, 6},
{4, -24, 24},
{0, 60, -180, 120},
{-24, 0, 720, -1440, 720},
{0, -840, 0, 8400, -12600, 5040},
{960, 0, -20160, 0, 100800, -120960, 40320},
{0, 60480, 0, -423360, 0, 1270080, -1270080, 362880},
{-120960, 0, 2419200, 0, -8467200, 0, 16934400, -14515200, 3628800},
{0, -11975040, 0, 79833600, 0, -167650560, 0, 239500800, -179625600, 39916800}, {36288000, 0, -718502400, 0, 2395008000, 0, -3353011200, 0, 3592512000, -2395008000, 479001600}
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MATHEMATICA
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p[t_]=t*GoldenRatio^(x*t)/(GoldenRatio^t-1); Table[ ExpandAll[((n+2)!*n!/Log[GoldenRatio]^(n-1))*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a=Table[ CoefficientList[((n+2)!*n!/Log[GoldenRatio]^(n-1))*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a] Table[Apply[Plus, CoefficientList[((n+2)!*n!/Log[GoldenRatio]^(n-1))*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x]], {n, 0, 10}];
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CROSSREFS
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Cf. A001898.
Sequence in context: A130879 A119741 A126063 this_sequence A096357 A091507 A034855
Adjacent sequences: A137521 A137522 A137523 this_sequence A137525 A137526 A137527
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KEYWORD
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tabl,sign
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 27 2008
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