A168641 Triangle read by rows: T(n,k) = [x^k] p(x,n), where p(x,n) = 3*(x + 1)^n - 2*(x^n + 1) - n*(x + x^(n - 1)) for n >= 2, p(x,0) = 1, and p(x,1) = x + 1.

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 8, 18, 8, 1, 1, 10, 30, 30, 10, 1, 1, 12, 45, 60, 45, 12, 1, 1, 14, 63, 105, 105, 63, 14, 1, 1, 16, 84, 168, 210, 168, 84, 16, 1, 1, 18, 108, 252, 378, 378, 252, 108, 18, 1, 1, 20, 135, 360, 630, 756, 630, 360, 135, 20, 1

Offset: 0

Author

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

Keywords

• nonn
• easy
• less
• tabl

Examples

			Triangle begins:
  1;
  1,  1;
  1,  2,   1;
  1,  6,   6,   1;
  1,  8,  18,   8,   1;
  1, 10,  30,  30,  10,   1;
  1, 12,  45,  60,  45,  12,   1;
  1, 14,  63, 105, 105,  63,  14,   1;
  1, 16,  84, 168, 210, 168,  84,  16,  1;
  1, 18, 108, 252, 378, 378, 252, 108,  18,  1;
  1, 20, 135, 360, 630, 756, 630, 360, 135, 20, 1;
  ...
		

Links

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

Crossrefs

Cf. A095151, A132046, A168643, A168644, A168646.

Columns (essentially): A005843 (k=1), A045943 (k=2), A027480 (k=3), A050534 (k=4), A253942 (k=5), A253943 (k=6), A253944 (k=7).

Programs

Magma

function f(n,k)
   if n le 2 then return 1;
   elif k eq 0 or k eq n then return 1;
   elif k eq 1 or k eq n-1 then return 2;
   else return 3;
   end if;
end function;
A168641:= func< n,k | Binomial(n,k)*f(n,k) >;
[A168641(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 24 2025

Mathematica

p[x_, n_]:= If[n==0, 1, If[n==1, 1+x, 3*(1+x)^n -(1+x^n) -(1+n*x +n*x^(n-1) + x^n)]];
Flatten[Table[CoefficientList[p[x, n], x], {n, 0, 10}]]
(* Second program *)
f[n_, k_]:= With[{b=Boole}, If[k<=n/2, b[k==0] +2*b[k==1] +3*b[2<=k<=n/2], f[n, n-k]]];
A168641[n_, k_]:= Binomial[n,k]*If[n<3,1,f[n,k]];
Table[A168641[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 24 2025 *)

Maxima

T(n,k) := ratcoef(if n <= 2 then (1 + x)^n else 3*(x + 1)^n - (x^n + 1) - (x^n + n*x^(n - 1) + n*x + 1), 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<=n/2): return int(k==0) + 2*int(k==1) + 3*int(1A168641(n,k):
    if (n<3): return binomial(n,k)
    else: return binomial(n,k)*f(n,k)
print(flatten([[A168641(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Mar 24 2025

Formula

From G. C. Greubel, Mar 24 2025: (Start)

T(n, k) = 3*binomial(n, k), for n >= 4 and 2 <= k <= n-2, otherwise T(n, 0) = T(n, n) = 1, T(n, 1) = T(n, n-1) = 2*A065475(n-1).

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

T(n, 1) = A005843(n) - [n=1] - 2*[n=2].

Columns: T(n, k) = 3*binomial(n,k) - 2*[n=k] - (k+1)*[n=k+1], k >= 2.

Sum_{k=0..n} T(n, k) = 2*A095151(n-1) - 2*[n=0] - 2*[n=1].

Sum_{k=0..n} (-1)^k*T(n, k) = (1+(-1)^n)*(n-2) + 5*[n=0]. (End)

Extensions

Edited by Franck Maminirina Ramaharo, Jan 02 2019