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 A137524 Triangular sequence from coefficients of the umbral calculus expansion of a Golden -Mean Bernoulli function(A001898): p(x,t)=t*phi^(x*t)/(phi^t - 1), where the golden ratio replaces "e". 0
 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; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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(phi)^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}]. LINKS Table of n, a(n) for n=1..52. EXAMPLE {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} MATHEMATICA 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}]; CROSSREFS Cf. A001898. Sequence in context: A214352 A248090 A229774 * A282507 A156055 A096357 Adjacent sequences: A137521 A137522 A137523 * A137525 A137526 A137527 KEYWORD tabl,sign AUTHOR Roger L. Bagula, Apr 27 2008 STATUS approved

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Last modified July 12 07:24 EDT 2024. Contains 374237 sequences. (Running on oeis4.)