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A132789 Triangle read by rows: T(n,k) = A007318(n-1, k-1) + A001263(n, k) - 1.

%I A132789 #10 Sep 08 2018 22:18:08
%S A132789 1,1,1,1,4,1,1,8,8,1,1,13,25,13,1,1,19,59,59,19,1,1,26,119,194,119,26,
%T A132789 1,1,34,216,524,524,216,34,1,1,43,363,1231,1833,1231,363,43,1,1,53,
%U A132789 575,2603,5417,5417,2603,575,53,1,1,64,869,5069,14069,19655,14069,5069,869
%N A132789 Triangle read by rows: T(n,k) = A007318(n-1, k-1) + A001263(n, k) - 1.
%H A132789 Andrew Howroyd, <a href="/A132789/b132789.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)
%F A132789 Equals A007318 + A001263 - A000012 as infinite lower triangular matrices.
%F A132789 A symmetrical triangle recursion: let q=4; 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).
%F A132789 T(n,k) = binomial(n-1, k-1)*(1 + binomial(n, k-1)/k) - 1. - _Andrew Howroyd_, Sep 08 2018
%e A132789 First few rows of the triangle are:
%e A132789   1;
%e A132789   1,  1;
%e A132789   1,  4,   1;
%e A132789   1,  8,   8,    1;
%e A132789   1, 13,  25,   13,     1;
%e A132789   1, 19,  59,   59,    19,     1;
%e A132789   1, 26, 119,  194,   119,    26,     1;
%e A132789   1, 34, 216,  524,   524,   216,    34,    1;
%e A132789   1, 43, 363, 1231,  1833,  1231,   363,   43,   1;
%e A132789   1, 53, 575, 2603,  5417,  5417,  2603,  575,  53,  1;
%e A132789   1, 64, 869, 5069, 14069, 19655, 14069, 5069, 869, 64, 1;
%e A132789   ...
%t A132789 << DiscreteMath`Combinatorica`
%t A132789 t[n_, m_, 0] := Binomial[n, m];
%t A132789 t[n_, m_, 1] := Binomial[n, m]*Binomial[n + 1, m]/(m + 1);
%t A132789 t[n_, m_, 2] := Eulerian[1 + n, m];
%t A132789 t[n_, m_, q_] := t[n, m, q] = t[n, m, q - 2] + t[n, m, q - 3] - 1;
%t A132789 Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 0, 10}]
%o A132789 (PARI) T(n,k)={if(k<=n, binomial(n-1, k-1)*(1 + binomial(n, k-1)/k) - 1, 0)}
%o A132789 for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print); \\ _Andrew Howroyd_, Sep 08 2018
%Y A132789 Column k=2 is A034856.
%Y A132789 Row sums are A132790.
%Y A132789 Cf. A007318, A001263.
%K A132789 nonn,tabl
%O A132789 1,5
%A A132789 _Gary W. Adamson_, Aug 30 2007
%E A132789 More terms, Mma program and additional comments from _Roger L. Bagula_, Apr 20 2010
%E A132789 Edited by _N. J. A. Sloane_, Apr 21 2010 at the suggestion of R. J. Mathar
%E A132789 Name clarified by _Andrew Howroyd_, Sep 08 2018