A168623 Triangle read by rows: T(n, k) = [x^k]( 9*(1+x)^n - 8*(1 + x^n) ), with T(0, 0) = 1.

1, 1, 1, 1, 18, 1, 1, 27, 27, 1, 1, 36, 54, 36, 1, 1, 45, 90, 90, 45, 1, 1, 54, 135, 180, 135, 54, 1, 1, 63, 189, 315, 315, 189, 63, 1, 1, 72, 252, 504, 630, 504, 252, 72, 1, 1, 81, 324, 756, 1134, 1134, 756, 324, 81, 1, 1, 90, 405, 1080, 1890, 2268, 1890, 1080, 405, 90, 1

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Dec 01 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  1,  1;
  1, 18,   1;
  1, 27,  27,    1;
  1, 36,  54,   36,    1;
  1, 45,  90,   90,   45,    1;
  1, 54, 135,  180,  135,   54,    1;
  1, 63, 189,  315,  315,  189,   63,    1;
  1, 72, 252,  504,  630,  504,  252,   72,   1;
  1, 81, 324,  756, 1134, 1134,  756,  324,  81,  1;
  1, 90, 405, 1080, 1890, 2268, 1890, 1080, 405, 90, 1;
		

Links

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

Crossrefs

Cf. A000045, A132047.

Programs

Magma

A168623:= func< n,k | k eq 0 or k eq n select 1 else 9*Binomial(n,k) >;
[A168623(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 09 2025

Mathematica

(* First program *)
p[x_, n_]:= With[{m=4}, If[n==0, 1, (2*m+1)(1+x)^n - 2*m*(1+x^n)]];
Table[CoefficientList[p[x,n], x], {n,0,10}]//Flatten
(* Second program *)
A168623[n_, k_]:= If[k==0 || k==n, 1, 9*Binomial[n,k]];
Table[A168623[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 09 2025 *)

SageMath

def A168623(n,k):
    if (k==0 or k==n): return 1
    else: return 9*binomial(n,k)
print(flatten([[A168623(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Apr 09 2025

Formula

From G. C. Greubel, Apr 09 2025: (Start)

T(n, k) = binomial(n,k)*( [k=0] + 9*[0 < k < n] + [k=n] ).

T(n, n-k) = T(n, k).

Sum_{k=0..n} T(n, k) = 9*2^n - 16 + 8*[n=0].

Sum_{k=0..n} (-1)^k*T(n, k) = -8*(1 + (-1)^n) + 17*[n=0].

Sum_{k=0..floor(n/2)} T(n-k, k) = 9*A000045(n+1) - 4*(3 + (-1)^n) + 8*[n=0]. (End)

Extensions

Keyword:tabl and row sum formula added - The Assoc. Editors of the OEIS, Dec 05 2009