login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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