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

0,5

Links

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

Formula

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

Example

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;
  ...

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}]]
(* Alternative: *)
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 *)

Prog

(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 0<n<7 and 0<k<n: return 8-n
    else: return (k+1)*int(k<5) + 6*int(4<k<n-4) + (n-k+1)*int(k>n-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

Crossrefs

Cf. A132046, A168641, A168643, A168644.

Sequence in context: A010206 A174096 A070636 * A051457 A174450 A166343

Adjacent sequences: A168643 A168644 A168645 * A168647 A168648 A168649

Keyword

nonn,tabl,easy,less

Author

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

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

Status

approved