A135227 Triangle A000012 * A135225, read by rows.

1, 2, 1, 3, 2, 1, 4, 3, 3, 1, 5, 4, 6, 4, 1, 6, 5, 10, 10, 5, 1, 7, 6, 15, 20, 15, 6, 1, 8, 7, 21, 35, 35, 21, 7, 1, 9, 8, 28, 56, 70, 56, 28, 8, 1, 10, 9, 36, 84, 126, 126, 84, 36, 9, 1, 11, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 12, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1

Offset

0,2

Comments

Row sums = A006127: (1, 3, 6, 11, 20, 37, ...).

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

A000012 * A135225 as infinite lower triangular matrices. Left border of 1's in Pascal's Triangle (A007318) is replaced with a column of (1,2,3,...).

T(n,k) = binomial(n,k), with T(n,0) = n+1. - G. C. Greubel, Nov 20 2019

Example

First few rows of the triangle:
  1;
  2, 1;
  3, 2,  1;
  4, 3,  3,  1;
  5, 4,  6,  4,  1;
  6, 5, 10, 10,  5, 1;
  7, 6, 15, 20, 15, 6, 1;
...

Maple

seq(seq( `if`(k=0, n+1, binomial(n, k)), k=0..n), n=0..12); # G. C. Greubel, Nov 20 2019

Mathematica

Table[If[k==0, n+1, Binomial[n, k]], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

Prog

(PARI) T(n, k) = if(k==0, n+1, binomial(n, k)); \\ G. C. Greubel, Nov 20 2019
(Magma) [k eq 0 select n+1 else Binomial(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2019
(Sage)
def T(n, k):
    if (k==0): return 1
    else: return binomial(n, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019
(GAP)
T:= function(n, k)
    if k=0 then return 1;
    else return Binomial(n, k);
    fi; end;
Flat(List([0..12], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Nov 20 2019

Crossrefs

Cf. A006127, A007318, A135225.

Sequence in context: A255238 A212536 A188277 * A104325 A204925 A369878

Adjacent sequences: A135224 A135225 A135226 * A135228 A135229 A135230

Keyword

nonn,tabl

Author

Gary W. Adamson, Nov 23 2007

Extensions

More terms added by G. C. Greubel, Nov 20 2019

Status

approved