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A symmetrical triangle sequence based on:q=2/12;t(n,m,q)=12*(Binomial[n, m]*(1 - q) + (((n + m + 1)!/(n + 1)!* m!) + ((2*n - m + 1)!/(n + 1)!*(n - m)!))*q)
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%I #2 Mar 30 2012 17:34:40

%S 1,1,1,1,-55,1,1,-2371,-2371,1,1,-141079,-144767,-141079,1,1,

%T -13063627,-13299239,-13299239,-13063627,1,1,-1756339135,-1778768213,

%U -1779131331,-1778768213,-1756339135,1,1,-323805081523,-326881525841,-326918009541

%N A symmetrical triangle sequence based on:q=2/12;t(n,m,q)=12*(Binomial[n, m]*(1 - q) + (((n + m + 1)!/(n + 1)!* m!) + ((2*n - m + 1)!/(n + 1)!*(n - m)!))*q)

%C Row sums are:

%C {1, 2, -53, -4740, -426923, -52725730, -8849346025, -1955209233808,

%C -552118682234375, -194345617889671998, -83514395998614084005,...}

%F q=2/12;

%F t(n,m,q)=12*(Binomial[n, m]*(1 - q) + (((n + m + 1)!/(n + 1)!* m!) + ((2*n - m + 1)!/(n + 1)!*(n - m)!))*q);

%F out_n,m,q=t(n,m,q)-t(n,0,q)+1

%e {1},

%e {1, 1},

%e {1, -55, 1},

%e {1, -2371, -2371, 1},

%e {1, -141079, -144767, -141079, 1},

%e {1, -13063627, -13299239, -13299239, -13063627, 1},

%e {1, -1756339135, -1778768213, -1779131331, -1778768213, -1756339135, 1},

%e {1, -323805081523, -326881525841, -326918009541, -326918009541, -326881525841, -323805081523, 1},

%e {1, -78460462079911, -79036461503291, -79041593014011, -79041649039951, -79041593014011, -79036461503291, -78460462079911, 1},

%e {1, -24186745110527899, -24328032061131923, -24329011779050579, -24329019994125599, -24329019994125599, -24329011779050579, -24328032061131923, -24186745110527899, 1},

%e {1, -9245027631071231887, -9289016465360485337, -9289260596277015803, -9289262200205068631, -9289262212786480691, -9289262200205068631, -9289260596277015803, -9289016465360485337, -9245027631071231887, 1}

%t t[n_, m_, q_] = 12*(Binomial[n, m]*(1 - q) + (((n + m + 1)!/(n + 1)!*m!) + ((2*n - m + 1)!/(n + 1)!*(n - m)!))*q);

%t Table[Flatten[Table[Table[t[ n, m, q] - t[n, 0, q] + 1, {m, 0, n}], {n, 0, 10}]], {q, 0, 1, 1/12}]

%K sign,tabl,uned

%O 0,5

%A _Roger L. Bagula_, Apr 02 2010