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

0,2

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

Links

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

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)

Example

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;

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 *)

Prog

(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

Crossrefs

Cf. A022114, A093644, A172179.

Sequence in context: A266703 A266701 A172283 * A098728 A233408 A107576

Adjacent sequences: A172182 A172183 A172184 * A172186 A172187 A172188

Keyword

nonn,tabl

Author

Mark Dols, Jan 28 2010

Extensions

Corrected and extended by Philippe Deléham, Oct 09 2011

Status

approved