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%I #2 Mar 30 2012 17:34:33
%S 1,1,1,1,10,1,3,42,42,3,3,128,262,128,3,5,340,1810,1810,340,5,7,958,
%T 8335,19326,8335,958,7,9,2468,38635,140569,140569,38635,2468,9,11,
%U 6022,160686,970572,1561898,970572,160686,6022,11,15,15193,669758,6372686
%N Triangle read by rows:e(n,k)=Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]; t(n,m)=(e[n + 1, m]*PartitionsQ[n] + e[n + 1, n - m]*(PartitionsQ[ n - m] + PartitionsQ[m])) - 2.
%C Row sums are:
%C {1, 2, 12, 90, 524, 4310, 37926, 363362, 3836480, 48184504, 643393254,...}.
%F e(n,k)=Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}];
%F t(n,m)=(e[n + 1, m]*PartitionsQ[n] + e[n + 1, n - m]*(PartitionsQ[ n - m] + PartitionsQ[m])) - 2.
%e {1},
%e {1, 1},
%e {1, 10, 1},
%e {3, 42, 42, 3},
%e {3, 128, 262, 128, 3},
%e {5, 340, 1810, 1810, 340, 5},
%e {7, 958, 8335, 19326, 8335, 958, 7},
%e {9, 2468, 38635, 140569, 140569, 38635, 2468, 9},
%e {11, 6022, 160686, 970572, 1561898, 970572, 160686, 6022, 11},
%e {15, 15193, 669758, 6372686, 17034600, 17034600, 6372686, 669758, 15193, 15},
%e {19, 38682, 2594827, 37459294, 155809822, 251587966, 155809822, 37459294, 2594827, 38682, 19}
%t Clear[e, k, t, n, m];
%t e[n_, k_] = Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}];
%t t[n_, m_] = (e[n + 1, m]*PartitionsQ[n] + e[n + 1, n - m]*( PartitionsQ[n - m] + PartitionsQ[m])) - 2;
%t Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t Flatten[%]
%K nonn,tabl,uned
%O 0,5
%A _Roger L. Bagula_, Feb 06 2009
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