A137369 - OEIS

A137369

Triangle read by rows: expansion of p(t) = (1 + t)^x/(1 + (1 + t)^n) with weight factor 2^(n+1)*n!.

%I #13 Feb 12 2024 08:41:01

%S 1,-1,2,4,-12,4,30,88,-60,8,-1344,224,752,-224,16,-16920,-31232,0,

%T 4320,-720,32,2977920,-430848,-371264,-10560,19840,-2112,64,53267760,

%U 104934912,-5789056,-3084928,-101920,78848,-5824,128,-24148131840,1882583040,1867684864,-54942720,-20344576,-645120,283136

%N Triangle read by rows: expansion of p(t) = (1 + t)^x/(1 + (1 + t)^n) with weight factor 2^(n+1)*n!.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BoolePolynomial.html">Boole Polynomial</a>.

%e {1},

%e {-1, 2},

%e {4, -12, 4},

%e {30, 88, -60, 8},

%e {-1344, 224, 752, -224, 16},

%e {-16920, -31232,0, 4320, -720, 32},

%e {2977920, -430848, -371264, -10560, 19840, -2112, 64},

%e {53267760, 104934912, -5789056, -3084928, -101920, 78848, -5824, 128},

%t p[t_] = (1 + t)^x/(1 + (1 + t)^n)

%t Table[ ExpandAll[2^(n + 1)*n!*SeriesCoefficient[Series[p[t], {t, 0, 30}], n]], {n, 0, 10}];

%t a = Table[ CoefficientList[2^(n + 1)*n!*SeriesCoefficient[Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}];

%t Flatten[a]

%K uned,tabl,sign

%O 1,3

%A _Roger L. Bagula_, Apr 09 2008