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

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

Offset: 1

Author

Gary W. Adamson, Aug 30 2007

Keywords

• nonn
• tabl

Examples

			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;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Crossrefs

Column k=2 is A034856.

Row sums are A132790.

Cf. A007318, A001263.

Programs

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}]

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

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

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