A124928 Triangle read by rows: T(n,0) = 1, T(n,k) = 3*binomial(n,k) if k>=0 (0<=k<=n).

1, 1, 3, 1, 6, 3, 1, 9, 9, 3, 1, 12, 18, 12, 3, 1, 15, 30, 30, 15, 3, 1, 18, 45, 60, 45, 18, 3, 1, 21, 63, 105, 105, 63, 21, 3, 1, 24, 84, 168, 210, 168, 84, 24, 3, 1, 27, 108, 252, 378, 378, 252, 108, 27, 3, 1, 30, 135, 360, 630, 756, 630, 360, 135, 30, 3

Offset

0,3

Comments

Row sums = A033484: (1, 4, 10, 22, 46, 94...); 3*2^n - 2.

Analogous triangle using (1,2,2,2...) as the main diagonal of M = A124927.

Except for the first column, entries in the Pascal triangle are tripled.

Links

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

Formula

G.f.: G(t,z) = 3/(1-(1+t)*z) - 2/(1-z).

Example

First few rows of the triangle are:
  1;
  1,  3;
  1,  6,  3;
  1,  9,  9,  3;
  1, 12, 18, 12,  3;
  1, 15, 30, 30, 15,  3;
  1, 18, 45, 60, 45, 18, 3;
...

Maple

T:=proc(n, k) if k=0 then 1 else 3*binomial(n, k) fi end: for n from 0 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form

Mathematica

Flatten[Table[If[k==0, 1, 3*Binomial[n, k]], {n, 0, 20}, {k, 0, n}]] (* Harvey P. Dale, Oct 19 2013 *)

Prog

(PARI) T(n, k) = if(k==0, 1, 3*binomial(n, k)); \\ G. C. Greubel, Nov 19 2019
(Magma) [k eq 0 select 1 else 3*Binomial(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 19 2019
(Sage)
def T(n, k):
    if (k==0): return 1
    else: return 3*binomial(n, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 19 2019
(GAP)
T:= function(n, k)
    if k=0 then return 1;
    else return 3*Binomial(n, k);
    fi;  end;
Flat(List([0..12], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Nov 19 2019

Crossrefs

Cf. A033484, A124927.

Sequence in context: A341219 A145366 A145367 * A249250 A122432 A131110

Adjacent sequences: A124925 A124926 A124927 * A124929 A124930 A124931

Keyword

nonn,tabl

Author

Gary W. Adamson, Nov 12 2006

Extensions

Edited by N. J. A. Sloane, Nov 29 2006

Status

approved