A135223 Triangle A000012 * A127648 * A103451, read by rows.

1, 3, 2, 6, 2, 3, 10, 2, 3, 4, 15, 2, 3, 4, 5, 21, 2, 3, 4, 5, 6, 28, 2, 3, 4, 5, 6, 7, 36, 2, 3, 4, 5, 6, 7, 8, 45, 2, 3, 4, 5, 6, 7, 8, 9, 55, 2, 3, 4, 5, 6, 7, 8, 9, 10, 66, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 78, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 91, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13

Offset: 1

Author

Gary W. Adamson, Nov 23 2007

Keywords

• nonn
• tabl

Comments

Row sums = A028387.

Examples

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

Links

• G. C. Greubel, Rows n = 1..100 of triangle, flattened

Crossrefs

Cf. A002260, A103451, A127648.

Programs

GAP

T:= function(n,k)
    if k=1 then return Binomial(n+1,2);
    else return k;
    fi; end;
Flat(List([1..15], n-> List([1..n], k-> T(n,k) ))); # G. C. Greubel, Nov 20 2019

Magma

[k eq 1 select Binomial(n+1,2) else k: k in [1..n], n in [1..15]]; // G. C. Greubel, Nov 20 2019

Maple

seq(seq( `if`(k=1, binomial(n+1,2), k), k=1..n), n=1..15); # G. C. Greubel, Nov 20 2019

Mathematica

T[n_, k_]:= T[n, k]= If[k==1, Binomial[n+1, 2], k]; Table[T[n, k], {n, 15}, {k,n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

PARI

T(n,k) = if(k==1, binomial(n+1,2), k); \\ G. C. Greubel, Nov 20 2019

Sage

@CachedFunction
def T(n,k):
    if (k==1): return binomial(n+1, 2)
    else: return k
[[T(n,k) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Nov 20 2019

Formula

T(n,k) = A000012(n,k) * A127648(n,k) * A103451(n,k) as infinite lower triangular matrices. Replace left border of 1's in A002260 with (1, 3, 6, 10, 15, ...).

T(n, k) = k with T(n,1) = binomial(n+1, 2). - G. C. Greubel, Nov 20 2019

Extensions

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