A131114 T(n,k) = 6*binomial(n,k) - 5*I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

1, 6, 1, 6, 12, 1, 6, 18, 18, 1, 6, 24, 36, 24, 1, 6, 30, 60, 60, 30, 1, 6, 36, 90, 120, 90, 36, 1, 6, 42, 126, 210, 210, 126, 42, 1, 6, 48, 168, 336, 420, 336, 168, 48, 1, 6, 54, 216, 504, 756, 756, 504, 216, 54, 1

Offset

0,2

Comments

Row sums give A048488.

Links

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

Formula

T(n,k) = 6*A007318(n,k) - 5*I(n,k), where A007318 = Pascal's triangle and I = Identity matrix.

Bivariate o.g.f.: Sum_{n,k>=0} T(n,k)*x^n*y^k = (1 + 5*x - x*y)/((1 - x*y)*(1 - x - x*y)).

Example

Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  6,  1;
  6, 12,  1;
  6, 18, 18,   1;
  6, 24, 36,  24,  1;
  6, 30, 60,  60, 30,  1;
  6, 36, 90, 120, 90, 36, 1;
  ...

Maple

seq(seq(`if`(k=n, 1, 6*binomial(n, k)), k=0..n), n=0..10); # G. C. Greubel, Nov 18 2019

Mathematica

Table[If[k==n, 1, 6*Binomial[n, k]], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)

Prog

(PARI) T(n, k) = if(k==n, 1, 6*binomial(n, k)); \\ G. C. Greubel, Nov 18 2019
(Magma) [k eq n select 1 else 6*Binomial(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 18 2019
(Sage)
def T(n, k):
    if (k==n): return 1
    else: return 6*binomial(n, k)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 18 2019
(GAP)
T:= function(n, k)
    if k=n then return 1;
    else return 6*Binomial(n, k);
    fi;  end;
Flat(List([0..10], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Nov 18 2019

Crossrefs

Cf. A007318, A048488, A131110, A131111, A131112, A131113, A131115.

Sequence in context: A272055 A157292 A159828 * A199230 A199101 A366868

Adjacent sequences: A131111 A131112 A131113 * A131115 A131116 A131117

Keyword

nonn,tabl,easy,less

Author

Gary W. Adamson, Jun 15 2007

Status

approved