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A symmetrical triangular sequence:t(n,m)=n!*(StirlingS1[n, m] + StirlingS1[n, n - m] - (StirlingS1[n, 0] + StirlingS1[n, n]) + 1) - n! + 1
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%I #2 Mar 30 2012 17:34:39

%S 1,1,1,1,-5,1,1,-11,-11,1,1,-311,505,-311,1,1,1561,-1919,-1919,1561,1,

%T 1,-97919,257761,-324719,257761,-97919,1,1,3517921,-8013599,4475521,

%U 4475521,-8013599,3517921,1,1,-204382079,539844481,-608549759,545811841

%N A symmetrical triangular sequence:t(n,m)=n!*(StirlingS1[n, m] + StirlingS1[n, n - m] - (StirlingS1[n, 0] + StirlingS1[n, n]) + 1) - n! + 1

%C Row sums are:

%C {1, 2, -3, -20, -115, -714, -5033, -40312, -362871, -3628790, -39916789,...}.

%F t(n,m)=n!*(StirlingS1[n, m] + StirlingS1[n, n - m] - (StirlingS1[n, 0] + StirlingS1[n, n]) + 1) - n! + 1

%e {1},

%e {1, 1},

%e {1, -5, 1},

%e {1, -11, -11, 1},

%e {1, -311, 505, -311, 1},

%e {1, 1561, -1919, -1919, 1561, 1},

%e {1, -97919, 257761, -324719, 257761, -97919, 1},

%e {1, 3517921, -8013599, 4475521, 4475521, -8013599, 3517921, 1},

%e {1, -204382079, 539844481, -608549759, 545811841, -608549759, 539844481, -204382079, 1},

%e {1, 14617895041, -39568072319, 41218450561, -16270087679, -16270087679, 41218450561, -39568072319, 14617895041, 1},

%e {1, -1316985868799, 3728392416001, -4289789548799, 2855691417601, -1954656748799, 2855691417601, -4289789548799, 3728392416001, -1316985868799, 1}

%t t[n_, m_] = n!*(StirlingS1[n, m] + StirlingS1[n, n - m] - (StirlingS1[ n, 0] + StirlingS1[n, n]) + 1) - n! + 1;

%t Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];

%t Flatten[%]

%K sign,tabl,uned

%O 0,5

%A _Roger L. Bagula_, Mar 31 2010