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A155467 Triangle T(n, k) = Eulerian(n+1, k)*Binomial(n+1, k)/(k+1), read by rows.

%I A155467 #9 Apr 03 2022 09:55:40
%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 1852914,3935988,1852914,175296,2259,1,1,5065,717600,13655760,55034868,55034868,13655760,717600,5065,1
%N A155467 Triangle T(n, k) = Eulerian(n+1, k)*Binomial(n+1, k)/(k+1), read by rows.
%C A155467 The sequence substitutes Eulerian numbers for the binomial in a triangle of Narayana numbers A001263,
%H A155467 G. C. Greubel, <a href="/A155467/b155467.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A155467 T(n, k) = binomial(n+1, k)*t(n, k, m)/(k+1), where t(n,k,m) = (m*(n-k)+1)*t(n-1,k-1,m) + (m*k-m+1)*t(n-1,k,m), t(n,1,m) = t(n,n,m) = 1, and m = 1.
%F A155467 Sum_{k=0..n} T(n, k) = A099765(n+2).
%F A155467 T(n, k) = Eulerian(n+1, k)*Binomial(n+1, k)/(k+1). - _Roger L. Bagula_, Apr 14 2010
%F A155467 From _G. C. Greubel_, Apr 01 2022: (Start)
%F A155467 T(n, k) = binomial(n+1, k)*A008292(n+1, k+1)/(k+1).
%F A155467 T(n, n-k) = T(n, k).
%F A155467 T(n, 1) = A003469(n). (End)
%e A155467 Triangle begins as:
%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;
%t A155467 (* First program *)
%t A155467 Needs["Combinatorica`"]
%t A155467 T[n_, k_]:= Eulerian[n+1, k]*Binomial[n+1, k]/(k+1);
%t A155467 Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* _Roger L. Bagula_, Apr 14 2010 *)
%t A155467 (* Second program *)
%t A155467 t[n_, k_, m_]:= t[n,k,m]= If[k==1 || k==n, 1, (m*n-m*k+1)*t[n-1,k-1,m] + (m*k -(m -1))*t[n-1,k,m]];
%t A155467 T[n_, k_, m_]:= Binomial[n+1,k]*t[n+1,k+1,m]/(k+1);
%t A155467 Table[T[n,k,1], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Apr 01 2022 *)
%o A155467 (Sage)
%o A155467 @CachedFunction
%o A155467 def t(n,k,m):
%o A155467     if (k==1 or k==n): return 1
%o A155467     else: return (m*(n-k)+1)*t(n-1,k-1,m) + (m*k-m+1)*t(n-1,k,m)
%o A155467 def T(n,k,m): return binomial(n+1,k)*t(n+1,k+1,m)/(k+1)
%o A155467 flatten([[T(n,k,1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 01 2022
%Y A155467 Cf. A001263 (m=0), this sequence (m=1), A155491 (m=3), A155493 (m=4).
%Y A155467 Cf. A001263, A008292, A099765 (row sums).
%K A155467 nonn,tabl
%O A155467 0,5
%A A155467 _Roger L. Bagula_, Jan 22 2009
%E A155467 Edited by _G. C. Greubel_, Apr 01 2022