A168643 Triangle read by rows: T(n,k) = [x^k] p(x,n) where p(x,0) = 1, p(x,n) = (6 - n)*(1+x)^n - (5-n)*(1 + x^n) for 1 <= n <= 4, and p (x,n) = 4*(1+x)^n - Sum_{i=0..2} (Sum_{j=0..i} binomial(n, j)*(x^j + x^(n-j))) for n >= 5.

1, 1, 1, 1, 8, 1, 1, 9, 9, 1, 1, 8, 12, 8, 1, 1, 10, 30, 30, 10, 1, 1, 12, 45, 80, 45, 12, 1, 1, 14, 63, 140, 140, 63, 14, 1, 1, 16, 84, 224, 280, 224, 84, 16, 1, 1, 18, 108, 336, 504, 504, 336, 108, 18, 1, 1, 20, 135, 480, 840, 1008, 840, 480, 135, 20, 1

Offset

0,5

Links

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

Formula

From G. C. Greubel, Apr 08 2025: (Start)

T(n, k) = [k=0] + (6-n)*binomial(n,k)*[1 <= k <= n-1] + [k=n] if 1 <= n <= 4, T(n, k) = binomial(n,k)*( (k+1)*[k<3] + 4*[2 < k < n-2] + (n-k+1)*[k > n-3] ) if n >= 5, with T(n, 0) = T(n, n) = 1.

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

Sum_{k=0..n} T(n, k) = 2^(n+2) - n^2 - 3*n - 6 + 13*[n=3] + 10*[n=2] + 4*[n=1] + 3*[n=0]. (End)

Example

Triangle begins:
  1;
  1,  1;
  1,  8,   1;
  1,  9,   9,   1;
  1,  8,  12,   8,   1;
  1, 10,  30,  30,  10,    1;
  1, 12,  45,  80,  45,   12,   1;
  1, 14,  63, 140, 140,   63,  14,   1;
  1, 16,  84, 224, 280,  224,  84,  16,   1;
  1, 18, 108, 336, 504,  504, 336, 108,  18,  1;
  1, 20, 135, 480, 840, 1008, 840, 480, 135, 20, 1;
  ...

Mathematica

(* First program *)
p[x_, n_]:= If[n==0, 1, If[n==1, x+1, 4*(1+x)^n - (1+x^n) - If[n>2, x^n + n*x^(n-1) +n*x+1, 1+x^n] - If[n>3, x^n +n*x^(n-1) + Binomial[n, 2]*(x^2 +x^(n-2)) +n*x+1, 1+x^n]]];
Flatten[Table[CoefficientList[p[x, n], x], {n, 0, 10}]]
(* Alternative: *)
f[n_, k_]:= With[{B=Boole}, If[n==0, 1, If[0<n<5, B[k==0] +(6-n)*B[0<k<n] +B[k==n], (k+1)*B[k<3] +4*B[2<k<n-2] +(n-k+1)*B[k>n-3]]]];
A168643[n_, k_]:= Binomial[n, k]*f[n, k];
Table[A168643[n, k], {n, 0, 13}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 08 2025 *)

Prog

(Maxima) T(n, k) := if k = 0 or k = n then 1 else (if n <= 4 then (6 - n)*binomial(n, k) else ratcoef(4*(x + 1)^n - sum(sum(binomial(n, j)*(x^j + x^(n - j)), j, 1, i), i, 1, 2), 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 n==0: return 1
    elif 0<n<5: return int(k==0) + (6-n)*int(0<k<n) + int(k==n)
    else: return (k+1)*int(k<3) + 4*int(2<k<n-2) + (n-k+1)*int(k>n-3)
def A168643(n, k): return binomial(n, k)*f(n, k)
print(flatten([[A168643(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Apr 08 2025

Crossrefs

Cf. A132046, A168641, A168644, A168646.

Sequence in context: A109571 A366494 A133823 * A173742 A146881 A174301

Adjacent sequences: A168640 A168641 A168642 * A168644 A168645 A168646

Keyword

nonn,easy,tabl,less

Author

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

Extensions

Edited by Franck Maminirina Ramaharo, Jan 02 2019

Status

approved