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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".

%I A137524 #5 Oct 01 2012 11:21:57
%S A137524 2,-3,6,4,-24,24,0,60,-180,120,-24,0,720,-1440,720,0,-840,0,8400,
%T A137524 -12600,5040,960,0,-20160,0,100800,-120960,40320,0,60480,0,-423360,0,
%U A137524 1270080,-1270080,362880,-120960,0,2419200,0,-8467200,0,16934400,-14515200,3628800,0,-11975040,0,79833600,0,-167650560,0
%N 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".
%C A137524 Row sums are: {2, 3, 4, 0, -24, 0, 960, 0, -120960, 0, 36288000}
%C A137524 These are the same as the Bernoulli numbers with the factor log(phi)^n: p[t_] = t*Exp[x*t]/(Exp[t] - 1);
%C A137524 a = Table[ CoefficientList[(n + 2)!*n!*SeriesCoefficient[Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}].
%e A137524 {2},
%e A137524 {-3, 6},
%e A137524 {4, -24, 24},
%e A137524 {0, 60, -180, 120},
%e A137524 {-24, 0, 720, -1440, 720},
%e A137524 {0, -840, 0, 8400, -12600, 5040},
%e A137524 {960, 0, -20160, 0, 100800, -120960, 40320},
%e A137524 {0, 60480, 0, -423360, 0, 1270080, -1270080, 362880},
%e A137524 {-120960, 0, 2419200, 0, -8467200, 0, 16934400, -14515200, 3628800},
%e A137524 {0, -11975040, 0, 79833600, 0, -167650560, 0, 239500800, -179625600, 39916800}, {36288000, 0, -718502400, 0, 2395008000, 0, -3353011200, 0, 3592512000, -2395008000, 479001600}
%t A137524 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}];
%Y A137524 Cf. A001898.
%K A137524 tabl,sign
%O A137524 1,1
%A A137524 _Roger L. Bagula_, Apr 27 2008