A172185 (9,11) Pascal triangle.

1, 9, 11, 9, 20, 11, 9, 29, 31, 11, 9, 38, 60, 42, 11, 9, 47, 98, 102, 53, 11, 9, 56, 145, 200, 155, 64, 11, 9, 65, 201, 345, 355, 219, 75, 11, 9, 74, 266, 546, 700, 574, 294, 86, 11, 9, 83, 340, 812, 1246, 1274, 868, 380, 97, 11, 9, 92, 423, 1152, 2058, 2520, 2142, 1248, 477, 108, 11

Offset: 0

Author

Mark Dols, Jan 28 2010

Keywords

• nonn
• tabl

Comments

Sums of NW-SE diagonals give A022114 (apart from first two terms).

Triangle T(n,k), read by rows, given by (9,-8,0,0,0,0,0,0,0,...) DELTA (11,-10,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 09 2011

Row n: Expansion of (9+11x)*(1+x)^(n-1), n > 0. - Philippe Deléham, Oct 09 2011

Examples

			Triangle begins:
  1;
  9, 11;
  9, 20,  11;
  9, 29,  31,   11;
  9, 38,  60,   42,   11;
  9, 47,  98,  102,   53,   11;
  9, 56, 145,  200,  155,   64,   11;
  9, 65, 201,  345,  355,  219,   75,   11;
  9, 74, 266,  546,  700,  574,  294,   86,  11;
  9, 83, 340,  812, 1246, 1274,  868,  380,  97,  11;
  9, 92, 423, 1152, 2058, 2520, 2142, 1248, 477, 108, 11;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A022114, A093644, A172179.

Programs

Mathematica

T[n_, k_]:= If[n==0, 1, (9 + 2*k/n)*Binomial[n, k]]
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 28 2022 *)

SageMath

def A172185(n,k): return 9*binomial(n,k) +2*binomial(n-1,k-1) -8*bool(n==0)
flatten([[A172185(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 28 2022

Formula

T(n,k) = T(n-1,k-1) + T(n-1,k) with T(0,0)=1, T(1,0)=9, T(1,1)=11. - Philippe Deléham, Oct 09 2011

G.f.: (1+8*x+10*y*x)/(1-x-y*x). - Philippe Deléham, Apr 13 2012

From G. C. Greubel, Apr 28 2022: (Start)

T(n, k) = 9*binomial(n, k) + 2*binomial(n-1, k-1) with T(0, 0) = 1.

Sum_{k=0..n} T(n, k) = 10*2^n - 9*[n=0]. (End)

Extensions

Corrected and extended by Philippe Deléham, Oct 09 2011