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

1,5

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: A334552 A347676 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