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A176560 A symmetrical triangle recursion:q=5;t(n,m,0)=Binomial[n,m];t(n,m,1)=Narayana(n,m);t(n,m,2)=Eulerian(n+1,m);t(n,m,q)=t(n,m,g-2)+t(n,m,q-3).

%I A176560 #4 Jun 02 2025 02:52:51
%S A176560 1,1,1,1,6,1,1,16,16,1,1,35,85,35,1,1,71,351,351,71,1,1,140,1295,2590,
%T A176560 1295,140,1,1,274,4488,16108,16108,4488,274,1,1,537,14943,89409,
%U A176560 157953,89409,14943,537,1,1,1057,48379,457711,1315645,1315645,457711,48379
%N A176560 A symmetrical triangle recursion:q=5;t(n,m,0)=Binomial[n,m];t(n,m,1)=Narayana(n,m);t(n,m,2)=Eulerian(n+1,m);t(n,m,q)=t(n,m,g-2)+t(n,m,q-3).
%C A176560 Row sums are:
%C A176560 {1, 2, 8, 34, 157, 846, 5462, 41742, 367733, 3645586, 39975575,...}.
%F A176560 q=5;
%F A176560 t(n,m,0)=Binomial[n,m];
%F A176560 t(n,m,1)=Narayana(n,m);
%F A176560 t(n,m,2)=Eulerian(n+1,m);
%F A176560 t(n,m,q)=t(n,m,g-2)+t(n,m,q-3)
%e A176560 {1},
%e A176560 {1, 1},
%e A176560 {1, 6, 1},
%e A176560 {1, 16, 16, 1},
%e A176560 {1, 35, 85, 35, 1},
%e A176560 {1, 71, 351, 351, 71, 1},
%e A176560 {1, 140, 1295, 2590, 1295, 140, 1},
%e A176560 {1, 274, 4488, 16108, 16108, 4488, 274, 1},
%e A176560 {1, 537, 14943, 89409, 157953, 89409, 14943, 537, 1},
%e A176560 {1, 1057, 48379, 457711, 1315645, 1315645, 457711, 48379, 1057, 1},
%e A176560 {1, 2090, 153461, 2208437, 9751973, 15743651, 9751973, 2208437, 153461, 2090, 1}
%t A176560 << DiscreteMath`Combinatorica`
%t A176560 t[n_, m_, 0] := Binomial[n, m];
%t A176560 t[n_, m_, 1] := Binomial[n, m]*Binomial[n + 1, m]/(m + 1);
%t A176560 t[n_, m_, 2] := Eulerian[1 + n, m];
%t A176560 t[n_, m_, q_] := t[n, m, q] = t[n, m, q - 2] + t[n, m, q - 3] - 1;
%t A176560 Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 0, 10}]
%Y A176560 Cf. A007318, A001263, A008292, A176490
%K A176560 nonn,tabl,uned
%O A176560 0,5
%A A176560 _Roger L. Bagula_, Apr 20 2010