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

1,2

Comments

Row sums = A028387.

Links

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

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

Example

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

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 *)

Prog

(PARI) T(n, k) = if(k==1, binomial(n+1, 2), 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
(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
(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

Crossrefs

Cf. A002260, A103451, A127648.

Sequence in context: A188614 A290798 A133519 * A143310 A131897 A061187

Adjacent sequences: A135220 A135221 A135222 * A135224 A135225 A135226

Keyword

nonn,tabl

Author

Gary W. Adamson, Nov 23 2007

Extensions

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

Status

approved