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".

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

Offset: 1

Author

Roger L. Bagula, Apr 27 2008

Keywords

• tabl
• sign

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}].

Examples

			{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}
		

Crossrefs

Cf. A001898.

Programs

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}];