A131115 Triangle read by rows: T(n,k) = 7*binomial(n,k) for 1 <= k <= n with T(n,n) = 1 for n >= 0.

1, 7, 1, 7, 14, 1, 7, 21, 21, 1, 7, 28, 42, 28, 1, 7, 35, 70, 70, 35, 1, 7, 42, 105, 140, 105, 42, 1, 7, 49, 147, 245, 245, 147, 49, 1, 7, 56, 196, 392, 490, 392, 196, 56, 1, 7, 63, 252, 588, 882, 882, 588, 252, 63, 1, 7, 70, 315, 840, 1470, 1764, 1470, 840, 315, 70, 1

Offset: 0

Author

Gary W. Adamson, Jun 15 2007

Keywords

• nonn
• tabl
• easy
• less

Comments

Row sums give A048489.

Non-diagonal entries of Pascal's triangle are multiplied by 7. - Emeric Deutsch, Jun 20 2007

The matrix inverse starts

1;

-7, 1;

91, -14, 1;

-1771, 273, -21, 1;

45955, -7084, 546, -28, 1;

-1490587, 229775, -17710, 910, -35, 1;

58018051, -8943522, 689325, -35420, 1365, -42, 1;

-2634606331, 406126357, -31302327, 1608425, -61985, 1911, -49, 1;

... - R. J. Mathar, Mar 15 2013

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  7,  1;
  7, 14,  1;
  7, 21, 21,  1;
  7, 28, 42, 28,  1;
  7, 35, 70, 70, 35, 1;
  ...
		

Links

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

Crossrefs

Cf. A007318, A048489, A131110, A131111, A131112, A131113, A131114.

Programs

GAP

T:= function(n,k)
    if k=n then return 1;
    else return 7*Binomial(n,k);
    fi;  end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 18 2019

Magma

[k eq n select 1 else 7*Binomial(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 18 2019

Maple

T := proc (n, k) if k < n then 7*binomial(n, k) elif k = n then 1 else 0 end if end proc; for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Jun 20 2007

Mathematica

Table[If[k==n, 1, 7*Binomial[n, k]], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)

PARI

T(n,k)=if(k==n,1,7*binomial(n,k)) \\ Charles R Greathouse IV, Jan 16 2012

Sage

@CachedFunction
def T(n, k):
    if (k==n): return 1
    else: return 7*binomial(n, k)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 18 2019

Formula

G.f.: (1 + 6*x - t*x)/((1-t*x)*(1-x-t*x)). - Emeric Deutsch, Jun 20 2007

Extensions

Corrected and extended by Emeric Deutsch, Jun 20 2007