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%I #2 Mar 30 2012 17:34:39
%S 1,1,1,1,4,1,1,16,16,1,1,69,99,69,1,1,318,548,548,318,1,1,1560,3024,
%T 3624,3024,1560,1,1,8139,17176,23161,23161,17176,8139,1,1,45094,
%U 101634,149374,168134,149374,101634,45094,1,1,264672,629226,989046,1214082
%N A triangular sequence:f(n)=Sum[StirlingS2[n, k], {k, 1, n}];t(n,m)=Binomial[n, m]*f(m + 1)*f(n - m + 1)-Binomial[n,0]*f(1)*f(n+1)+1
%C Row sums are:
%C 1, 2, 6, 34, 239, 1734, 12794, 96954, 760340, 6194054, 52490379,...
%F f(n)=Sum[StirlingS2[n, k], {k, 1, n}];
%F t(n,m)=Binomial[n, m]*f(m + 1)*f(n - m + 1)-Binomial[n,0]*f(1)*f(n+1)+1
%e {1},
%e {1, 1},
%e {1, 4, 1},
%e {1, 16, 16, 1},
%e {1, 69, 99, 69, 1},
%e {1, 318, 548, 548, 318, 1},
%e {1, 1560, 3024, 3624, 3024, 1560, 1},
%e {1, 8139, 17176, 23161, 23161, 17176, 8139, 1},
%e {1, 45094, 101634, 149374, 168134, 149374, 101634, 45094, 1},
%e {1, 264672, 629226, 989046, 1214082, 1214082, 989046, 629226, 264672, 1},
%e {1, 1640931, 4079506, 6773431, 8898271, 9706099, 8898271, 6773431, 4079506, 1640931, 1}
%t f[n_] := Sum[StirlingS2[n, k], {k, 1, n}];
%t t[n_, m_] = Binomial[n, m]*f[m + 1]*f[n - m + 1]
%t Table[Table[t[n, m] - t[n, 0] + 1, {m, 0, n}], {n, 0, 10}];
%t Flatten[%]
%K nonn,tabl,uned
%O 0,5
%A _Roger L. Bagula_, Mar 25 2010
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