A176263 Triangle T(n,k) = A015440(k) - A015440(n) + A015440(n-k), read by rows.

1, 1, 1, 1, -4, 1, 1, -4, -4, 1, 1, -29, -29, -29, 1, 1, -54, -79, -79, -54, 1, 1, -204, -254, -279, -254, -204, 1, 1, -479, -679, -729, -729, -679, -479, 1, 1, -1504, -1979, -2179, -2204, -2179, -1979, -1504, 1, 1, -3904, -5404, -5879, -6054, -6054, -5879, -5404, -3904, 1

Offset: 0

Author

Roger L. Bagula, Apr 13 2010

Keywords

• sign
• tabl
• easy

Comments

Row sums are s(n) = {1, 2, -2, -6, -85, -264, -1193, -3772, -13526, -42480, -139159, ...}, obeying s(n) = 3*s(n-1) + 7*s(n-2) - 19*s(n-3) - 15*s(n-4) + 25*s(n-5) with g.f. (1-x-15*x^2+5*x^3)/((1-x)*(1-x-5*x^2)^2).

Examples

			Triangle begins as:
  1;
  1,     1;
  1,    -4,     1;
  1,    -4,    -4,     1;
  1,   -29,   -29,   -29,     1;
  1,   -54,   -79,   -79,   -54,     1;
  1,  -204,  -254,  -279,  -254,  -204,     1;
  1,  -479,  -679,  -729,  -729,  -679,  -479,     1;
  1, -1504, -1979, -2179, -2204, -2179, -1979, -1504,     1;
  1, -3904, -5404, -5879, -6054, -6054, -5879, -5404, -3904, 1;
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Programs

Magma

A015440:= func< n | &+[5^j*Binomial(n-j,j): j in [0..Floor(n/2)]] >;
[A015440(k) - A015440(n) + A015440(n-k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 24 2019

Maple

A176263 := proc(n,k)
        A015440(k)-A015440(n)+A015440(n-k) ;
end proc; # R. J. Mathar, May 03 2013

Mathematica

(* Set of sequences q=0..10. This sequence is q=5. *)
f[n_, q_]:= f[n, q] = If[n<2, n, f[n-1, q] + q*f[n-2, q]];
T[n_, k_, q_]:= f[k+1, q] + f[n-k+1, q] - f[n+1, q];
Table[Flatten[Table[T[n, k, q], {n,0,10}, {k,0,n}]], {q,0,10}]
(* Second program *)
A015440[n_]:= Sum[5^j*Binomial[n-j, j], {j,0,(n+1)/2}]; T[n_, k_]:= T[n, k]= A015440[k] +A015440[n-k] -A015440[n]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 24 2019 *)

PARI

A015440(n) = sum(j=0,(n+1)\2, 5^j*binomial(n-j,j));
T(n,k) = A015440(k) - A015440(n) + A015440(n-k); \\ G. C. Greubel, Nov 24 2019

Sage

def A015440(n): return sum(5^j*binomial(n-j,j) for j in (0..floor(n/2)))
[[A015440(k) - A015440(n) + A015440(n-k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 24 2019