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A triangular sequence of the expansion of: (1-Prime[n])^n: t(n,m)=(-1)^m*Prime[n]^(n - m)*Binomial[n, m]; with Prime[0]=1 defined to extend and lower the results.
0

%I #2 Mar 30 2012 17:34:26

%S 1,2,-1,9,-6,1,125,-75,15,-1,2401,-1372,294,-28,1,161051,-73205,13310,

%T -1210,55,-1,4826809,-2227758,428415,-43940,2535,-78,1,410338673,

%U -168962983,29816997,-2923235,171955,-6069,119,-1,16983563041,-7150973912,1317284668,-138661544,9122470,-384104

%N A triangular sequence of the expansion of: (1-Prime[n])^n: t(n,m)=(-1)^m*Prime[n]^(n - m)*Binomial[n, m]; with Prime[0]=1 defined to extend and lower the results.

%C Row sums are:

%C {1, 1, 4, 64, 1296, 100000, 2985984, 268435456, 11019960576, 1207269217792,

%C 296196766695424}

%F t(n,m)=(-1)^m*Prime[n]^(n - m)*Binomial[n, m].

%e {1},

%e {2, -1},

%e {9, -6, 1},

%e {125, -75,15, -1},

%e {2401, -1372, 294, -28, 1},

%e {161051, -73205, 13310, -1210,55, -1},

%e {4826809, -2227758, 428415, -43940, 2535, -78, 1},

%e {410338673, -168962983, 29816997, -2923235, 171955, -6069, 119, -1},

%e {16983563041, -7150973912, 1317284668, -138661544,9122470, -384104, 10108, -152, 1},

%e {1801152661463, -704798867529, 122573716092, -12435014676, 810979218, -35259966, 1022028, -19044, 207, -1},

%e {420707233300201, -145071459758690, 22511088583245, -2069985157080, 124912897410, -5168809548, 148529010, -2926680, 37845, -290, 1}

%t Clear[T, n, m] T[n_, m_] = If[n == 0, (-1)^m*Binomial[n, m], (-1)^m*Prime[n]^(n - m)*Binomial[n, m]]; a = Table[Table[T[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[a]

%K uned,tabl,sign

%O 1,2

%A _Roger L. Bagula_, Jul 29 2008