A168622 Triangle read by rows: T(n, k) = [x^k]( 7*(1+x)^n - 6*(1+x^n) ) with T(0, 0) = 1.

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

Offset: 0

Author

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

Keywords

• nonn
• tabl
• easy

Examples

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

Links

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

Crossrefs

Cf. A022090, A131115, A132047, A168620, A168623.

Columns (essentially): A008589 (k=1), A024966 (k=2).

Programs

Magma

A168622:= func< n,k | k eq 0 or k eq n select 1 else 7*Binomial(n,k) >;
[A168622(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 10 2025

Mathematica

(* First program *)
p[x_, n_]:= With[{m=3}, If[n==0, 1, (2*m+1)(1+x)^n - 2*m*(1+x^n)]];
Table[CoefficientList[p[x,n], x], {n,0,12}]//Flatten
(* Second program *)
A168622[n_, k_]:= If[k==0 || k==n, 1, 7*Binomial[n,k]];
Table[A168622[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 10 2025 *)

SageMath

def A168622(n,k):
    if k==0 or k==n: return 1
    else: return 7*binomial(n,k)
print(flatten([[A168622(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Apr 10 2025

Formula

From G. C. Greubel, Apr 10 2025: (Start)

T(n, k) = 7*binomial(n, k), with T(n, 0) = T(n, n) = 1.

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

Sum_{k=0..n} T(n, k) = 2*A048489(n-1) + 6*[n=0].

Sum_{k=0..n} (-1)^k*T(n, k) = -6*(1 + (-1)^n) + 13*[n=0].

Sum_{k=0..floor(n/2)} T(n-k, k) = A022090(n+1) - 3*(3 + (-1)^n) + 6*[n=0].

Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (14/sqrt(3))*(-1)^n*cos((4*n+1)*Pi/6) - 6*(1 + (-1)^n*cos(n*Pi/2)) + 6*[n=0]. (End)