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A176561 A symmetrical triangle recursion:q=6;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 A176561 #4 Jun 02 2025 02:52:59
%S A176561 1,1,1,1,7,1,1,18,18,1,1,38,90,38,1,1,75,360,360,75,1,1,145,1309,2609,
%T A176561 1309,145,1,1,280,4508,16142,16142,4508,280,1,1,544,14970,89464,
%U A176561 158022,89464,14970,544,1,1,1065,48414,457794,1315770,1315770,457794,48414
%N A176561 A symmetrical triangle recursion:q=6;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 A176561 Row sums are:
%C A176561 {1, 2, 9, 38, 168, 872, 5519, 41862, 367980, 3646088, 39976588,...}.
%F A176561 q=6;
%F A176561 t(n,m,0)=Binomial[n,m];
%F A176561 t(n,m,1)=Narayana(n,m);
%F A176561 t(n,m,2)=Eulerian(n+1,m);
%F A176561 t(n,m,q)=t(n,m,g-2)+t(n,m,q-3)
%e A176561 {1},
%e A176561 {1, 1},
%e A176561 {1, 7, 1},
%e A176561 {1, 18, 18, 1},
%e A176561 {1, 38, 90, 38, 1},
%e A176561 {1, 75, 360, 360, 75, 1},
%e A176561 {1, 145, 1309, 2609, 1309, 145, 1},
%e A176561 {1, 280, 4508, 16142, 16142, 4508, 280, 1},
%e A176561 {1, 544, 14970, 89464, 158022, 89464, 14970, 544, 1},
%e A176561 {1, 1065, 48414, 457794, 1315770, 1315770, 457794, 48414, 1065, 1},
%e A176561 {1, 2099, 153505, 2208556, 9752182, 15743902, 9752182, 2208556, 153505, 2099, 1}
%t A176561 << DiscreteMath`Combinatorica`
%t A176561 t[n_, m_, 0] := Binomial[n, m];
%t A176561 t[n_, m_, 1] := Binomial[n, m]*Binomial[n + 1, m]/(m + 1);
%t A176561 t[n_, m_, 2] := Eulerian[1 + n, m];
%t A176561 t[n_, m_, q_] := t[n, m, q] = t[n, m, q - 2] + t[n, m, q - 3] - 1;
%t A176561 Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 0, 10}]
%Y A176561 Cf. A007318, A001263, A008292, A176490
%K A176561 nonn,tabl,uned
%O A176561 0,5
%A A176561 _Roger L. Bagula_, Apr 20 2010