A168646 Triangle read by rows: T(n,k) = [x^k] p(x,n), where p(x,0) = 1, p(x,n) = (8 - n)*(1+x)^n - (7 - n)*(1 + x^n) for 1 <= n <= 6, and p(x,n) = 6*(1+x)^n - Sum_{i=0..4} (Sum_{j=0..i} binomial(n, j)*(x^j + x^(n-j))) for n >= 7.

1, 1, 1, 1, 12, 1, 1, 15, 15, 1, 1, 16, 24, 16, 1, 1, 15, 30, 30, 15, 1, 1, 12, 30, 40, 30, 12, 1, 1, 14, 63, 315, 315, 63, 14, 1, 1, 16, 84, 224, 700, 224, 84, 16, 1, 1, 18, 108, 336, 630, 630, 336, 108, 18, 1, 1, 20, 135, 480, 1050, 1512, 1050, 480, 135, 20, 1, 1, 22, 165, 660, 1650, 2772, 2772, 1650, 660, 165, 22, 1

Offset: 0

Author

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

Keywords

• nonn
• tabl
• easy
• less

Examples

			Triangle begins:
  1;
  1,  1;
  1, 12,   1;
  1, 15,  15,   1;
  1, 16,  24,  16,    1;
  1, 15,  30,  30,   15,    1;
  1, 12,  30,  40,   30,   12,    1;
  1, 14,  63, 315,  315,   63,   14,   1;
  1, 16,  84, 224,  700,  224,   84,  16,   1;
  1, 18, 108, 336,  630,  630,  336, 108,  18,  1;
  1, 20, 135, 480, 1050, 1512, 1050, 480, 135, 20, 1;
  ...
		

Links

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

Crossrefs

Cf. A132046, A168641, A168643, A168644.

Programs

Mathematica

(* First program *)
p[n_, x_]:= With[{B=Binomial}, If[n==0, 1, If[1<=n<=6, 1 + (8-n)*Sum[B[n,j]*x^j, {j, n -1}] +x^n, Sum[(j+1)*B[n,j]*x^j, {j,0,4}] +6*Sum[B[n,j]*x^j, {j,5,n-5}] + Sum[(n-j+ 1)*B[n,j]*x^j, {j,n-4,n}]]]];
Flatten[Table[CoefficientList[p[n,x], x], {n, 0, 12}]]
(* Second program *)
f[n_, k_]:= If[k==0||k==n,1,If[1<=n<= 6 && 1<=k<=n-1, 8-n, (k+1)*Boole[k<=4] + 6*Boole[5<=k<=n-5] +(n-k+1)*Boole[n-4<=k<=n]]];
A168646[n_, k_]:= Binomial[n,k]*f[n,k];
Table[A168646[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 05 2025 *)

Maxima

T(n,k) := if k = 0 or k = n then 1 else (if n <= 6 then (8 - n)*binomial(n, k) else ratcoef(6*(x + 1)^n - sum(sum(binomial(n, j)*(x^j + x^(n - j)), j, 1, i), i, 1, 4), x, k))$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Jan 02 2019 */

SageMath

def f(n,k):
    if k==0 or k==n: return 1
    elif 0n-5)
def A168646(n,k): return binomial(n,k)*f(n,k)
print(flatten([[A168646(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Apr 05 2025

Formula

T(n, n-k) = T(n, k). - G. C. Greubel, Apr 05 2025

Extensions

Edited by Franck Maminirina Ramaharo, Jan 02 2019

Data values T(7,3), T(7,4), T(8,4) corrected by G. C. Greubel, Apr 05 2025