%I A155467
%S A155467 1,1,1,1,6,1,1,22,22,1,1,65,220,65,1,1,171,1510,1510,171,1,1,420,8337,
%T A155467 21140,8337,420,1,1,988,40068,218666,218666,40068,988,1,1,2259,175296,
%U A155467 1852914,3935988,1852914,175296,2259,1,1,5065,717600,13655760,55034868
%N A155467 Triangle read by rows: f(n)=(n+1)!;a(n,m)=f(n)/(f(m)*f(n-m)); e(n,m)=Sum[(-1)^j 
               Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]; t(n,m)=e(n,m)*a(n,
               m).
%C A155467 Row sums are:A099765;
%C A155467 {1, 2, 8, 46, 352, 3364, 38656, 519446, 7996928, 138826588, 2683604992,
               ...}.
%C A155467 The sequence substitutes Eulerian numbers for the binomial in a triangle 
               of
%C A155467 Narayana numbers A001263,
%F A155467 f(n)=(n+1)!;a(n,m)=f(n)/(f(m)*f(n-m));
%F A155467 e(n,m)=Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}];
%F A155467 t(n,m)=e(n,m)*a(n,m).
%e A155467 {1},
%e A155467 {1, 1},
%e A155467 {1, 6, 1},
%e A155467 {1, 22, 22, 1},
%e A155467 {1, 65, 220, 65, 1},
%e A155467 {1, 171, 1510, 1510, 171, 1},
%e A155467 {1, 420, 8337, 21140, 8337, 420, 1},
%e A155467 {1, 988, 40068, 218666, 218666, 40068, 988, 1},
%e A155467 {1, 2259, 175296, 1852914, 3935988, 1852914, 175296, 2259, 1},
%e A155467 {1, 5065, 717600, 13655760, 55034868, 55034868, 13655760, 717600, 5065, 
               1},
%e A155467 {1, 11198, 2798345, 90893880, 642715524, 1210767096, 642715524, 90893880, 
               2798345, 11198, 1}
%t A155467 t[n_, k_] = Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}];
%t A155467 f[n_] = Product[k + 1, {k, 0, n}];
%t A155467 A[n_, m_] = f[n]/(f[m]*f[n - m]);
%t A155467 Table[Table[t[n + 1, m]*A[n, m], {m, 0, n}], {n, 0, 10}];
%t A155467 Flatten[%]
%Y A155467 A001263, A099765
%Y A155467 Sequence in context: A056941 A157638 A142596 this_sequence A152936 A152969 
               A060187
%Y A155467 Adjacent sequences: A155464 A155465 A155466 this_sequence A155468 A155469 
               A155470
%K A155467 nonn,tabl,uned
%O A155467 0,5
%A A155467 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 22 2009