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 A132789 Triangle read by rows: T(n,k) = A007318(n-1, k-1) + A001263(n, k) - 1. 2
 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, 1, 1, 34, 216, 524, 524, 216, 34, 1, 1, 43, 363, 1231, 1833, 1231, 363, 43, 1, 1, 53, 575, 2603, 5417, 5417, 2603, 575, 53, 1, 1, 64, 869, 5069, 14069, 19655, 14069, 5069, 869 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 LINKS Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows) FORMULA Equals A007318 + A001263 - A000012 as infinite lower triangular matrices. 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). T(n,k) = binomial(n-1, k-1)*(1 + binomial(n, k-1)/k) - 1. - Andrew Howroyd, Sep 08 2018 EXAMPLE First few rows of the triangle are:   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,     1;   1, 34, 216,  524,   524,   216,    34,    1;   1, 43, 363, 1231,  1833,  1231,   363,   43,   1;   1, 53, 575, 2603,  5417,  5417,  2603,  575,  53,  1;   1, 64, 869, 5069, 14069, 19655, 14069, 5069, 869, 64, 1;   ... MATHEMATICA << DiscreteMath`Combinatorica` t[n_, m_, 0] := Binomial[n, m]; t[n_, m_, 1] := Binomial[n, m]*Binomial[n + 1, m]/(m + 1); t[n_, m_, 2] := Eulerian[1 + n, m]; t[n_, m_, q_] := t[n, m, q] = t[n, m, q - 2] + t[n, m, q - 3] - 1; Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 0, 10}] PROG (PARI) T(n, k)={if(k<=n, binomial(n-1, k-1)*(1 + binomial(n, k-1)/k) - 1, 0)} for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Sep 08 2018 CROSSREFS Column k=2 is A034856. Row sums are A132790. Cf. A007318, A001263. Sequence in context: A158687 A141541 A177947 * A319251 A100754 A332307 Adjacent sequences:  A132786 A132787 A132788 * A132790 A132791 A132792 KEYWORD nonn,tabl AUTHOR Gary W. Adamson, Aug 30 2007 EXTENSIONS More terms, Mma program and additional comments from Roger L. Bagula, Apr 20 2010 Edited by N. J. A. Sloane, Apr 21 2010 at the suggestion of R. J. Mathar Name clarified by Andrew Howroyd, Sep 08 2018 STATUS approved

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Last modified April 10 01:35 EDT 2020. Contains 333392 sequences. (Running on oeis4.)