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A triangle sequence derived from setting an Euler numbers A122045 generalization equal to the MacMahon numbers A060187 to get a generating function expansion: p(x,t) = (exp(t)* (1 - exp(x))* x)/(exp(2 t + t x) + exp(t)* x - exp(t*x)* x).
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%I #8 Dec 24 2017 10:31:43

%S 1,-1,-2,1,1,7,1,-7,2,-1,-22,-30,44,25,-30,6,1,69,244,-106,-495,193,

%T 242,-156,24,-1,-216,-1575,-1010,4695,1884,-5429,270,2190,-960,120,1,

%U 671,9251,19283,-28295,-59839,50065,56385,-52318,-9660,20640,-6840,720,-1

%N A triangle sequence derived from setting an Euler numbers A122045 generalization equal to the MacMahon numbers A060187 to get a generating function expansion: p(x,t) = (exp(t)* (1 - exp(x))* x)/(exp(2 t + t x) + exp(t)* x - exp(t*x)* x).

%C The equation solved for integer q is q*exp(x*t)/(q - 1 + exp(x)) - exp(t) *x/(-exp(2*t) + x) = 0.

%C The row sums are alternating sign powers of two: {1, -2, 4, -8, 16, -32, 64, -128, 256, -512, 1024, ...}.

%C This result comes out backward to the one in A178232 and took fewer factors to get the simple representation here.

%C There seems to be no q=2 lower limit/offset.

%F p(x,t) = (exp(t)* (1 - exp(x))* x)/(exp(2 t + t x) + exp(t)* x - exp(t*x)* x).

%e {1},

%e {-1, -2, 1},

%e {1, 7, 1, -7, 2},

%e {-1, -22, -30, 44, 25, -30, 6},

%e {1, 69, 244, -106, -495, 193, 242, -156, 24},

%e {-1, -216, -1575, -1010, 4695, 1884, -5429, 270, 2190, -960, 120},

%e {1, 671, 9251, 19283, -28295, -59839, 50065, 56385, -52318, -9660, 20640, -6840, 720},

%e {-1, -2066, -51772, -214172, 34412, 858928, 37982, -1387604, 227893, 1013586, -461874, -213360, 208320, -55440, 5040},

%e {1, 6313, 281590, 1998792, 2153998, -8664306, -10413872, 18525228, 14418817, -23553535, -5286622, 15734628, -3461976, -3543120, 2268000, -504000, 40320},

%e {-1, -19180, -1502997, -17111610, -43599684, 56619648, 230842878, -108741900, -482084361, 219258620, 495244735, -318866250, -213584610, 229612320, -12320280, -55671840, 26641440, -5080320, 362880},

%e {1, 58035, 7914157, 139210285, 615521460, 70210002, -3458014182, -1919690334, 8952377385, 3668682775, -13675554759, -1255675223, 12213830402, -2992953900, -5325088080, 3225783960, 328512240, -879379200, 336873600, -56246400, 3628800}

%t p[t_] = (Exp[t]* (1 - Exp[x])* x)/(Exp[2 t + t x] + Exp[t]* x - Exp[t*x]* x);

%t a = Table[ CoefficientList[FullSimplify[ExpandAll[(n!/( x*(1 - Exp[x])))*SeriesCoefficient[ Series[p[ t], {t, 0, 30}], n]]], x], {n, 0, 10}];

%t Flatten[a]

%Y Cf. A060187, A122045, A178232.

%K sign,tabf,uned

%O 0,3

%A _Roger L. Bagula_, May 23 2010