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 A178234 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). 0
 1, -1, -2, 1, 1, 7, 1, -7, 2, -1, -22, -30, 44, 25, -30, 6, 1, 69, 244, -106, -495, 193, 242, -156, 24, -1, -216, -1575, -1010, 4695, 1884, -5429, 270, 2190, -960, 120, 1, 671, 9251, 19283, -28295, -59839, 50065, 56385, -52318, -9660, 20640, -6840, 720, -1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The equation solved for integer q is q*exp(x*t)/(q - 1 + exp(x)) - exp(t) *x/(-exp(2*t) + x) = 0. The row sums are alternating sign powers of two: {1, -2, 4, -8, 16, -32, 64, -128, 256, -512, 1024, ...}. This result comes out backward to the one in A178232 and took fewer factors to get the simple representation here. There seems to be no q=2 lower limit/offset. LINKS FORMULA p(x,t) = (exp(t)* (1 - exp(x))* x)/(exp(2 t + t x) + exp(t)* x - exp(t*x)* x). EXAMPLE {1}, {-1, -2, 1}, {1, 7, 1, -7, 2}, {-1, -22, -30, 44, 25, -30, 6}, {1, 69, 244, -106, -495, 193, 242, -156, 24}, {-1, -216, -1575, -1010, 4695, 1884, -5429, 270, 2190, -960, 120}, {1, 671, 9251, 19283, -28295, -59839, 50065, 56385, -52318, -9660, 20640, -6840, 720}, {-1, -2066, -51772, -214172, 34412, 858928, 37982, -1387604, 227893, 1013586, -461874, -213360, 208320, -55440, 5040}, {1, 6313, 281590, 1998792, 2153998, -8664306, -10413872, 18525228, 14418817, -23553535, -5286622, 15734628, -3461976, -3543120, 2268000, -504000, 40320}, {-1, -19180, -1502997, -17111610, -43599684, 56619648, 230842878, -108741900, -482084361, 219258620, 495244735, -318866250, -213584610, 229612320, -12320280, -55671840, 26641440, -5080320, 362880}, {1, 58035, 7914157, 139210285, 615521460, 70210002, -3458014182, -1919690334, 8952377385, 3668682775, -13675554759, -1255675223, 12213830402, -2992953900, -5325088080, 3225783960, 328512240, -879379200, 336873600, -56246400, 3628800} MATHEMATICA p[t_] = (Exp[t]* (1 - Exp[x])* x)/(Exp[2 t + t x] + Exp[t]* x - Exp[t*x]* x); a = Table[ CoefficientList[FullSimplify[ExpandAll[(n!/( x*(1 - Exp[x])))*SeriesCoefficient[ Series[p[ t], {t, 0, 30}], n]]], x], {n, 0, 10}]; Flatten[a] CROSSREFS Cf. A060187, A122045, A178232. Sequence in context: A025270 A249450 A247450 * A259175 A297431 A301922 Adjacent sequences:  A178231 A178232 A178233 * A178235 A178236 A178237 KEYWORD sign,tabf,uned AUTHOR Roger L. Bagula, May 23 2010 STATUS approved

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Last modified October 17 15:01 EDT 2019. Contains 328116 sequences. (Running on oeis4.)