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).

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

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

Table of n, a(n) for n=0..49.

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: A373343 A331501 A247450 * A344440 A259175 A297431

Adjacent sequences: A178231 A178232 A178233 * A178235 A178236 A178237

Keyword

sign,tabf,uned

Author

Roger L. Bagula, May 23 2010

Status

approved