A176264 Triangle T(n,k) = A015442(k) - A015442(n) + A015442(n-k) read by rows.

1, 1, 1, 1, -6, 1, 1, -6, -6, 1, 1, -55, -55, -55, 1, 1, -104, -153, -153, -104, 1, 1, -496, -594, -643, -594, -496, 1, 1, -1231, -1721, -1819, -1819, -1721, -1231, 1, 1, -4710, -5935, -6425, -6474, -6425, -5935, -4710, 1, 1, -13334, -18038, -19263, -19704, -19704, -19263, -18038, -13334, 1

Offset: 0

Author

Roger L. Bagula, Apr 13 2010

Keywords

• sign
• tabl
• easy

Comments

Row sums are s(n) = {1, 2, -4, -10, -163, -512, -2821, -9540, -40612, -140676, -537533, ...} where s(n) = 3*s(n-1) +11*s(n-2) -27*s(n-3) -35*s(n-4) +49*s(n-5) with g.f. (1-x-21*x^2+7*x^3)/((1-x)*(1-x-7*x^2)^2).

Examples

			Triangle begins as:
  1;
  1,      1;
  1,     -6,      1;
  1,     -6,     -6,      1;
  1,    -55,    -55,    -55,      1;
  1,   -104,   -153,   -153,   -104,      1;
  1,   -496,   -594,   -643,   -594,   -496,      1;
  1,  -1231,  -1721,  -1819,  -1819,  -1721,  -1231,      1;
  1,  -4710,  -5935,  -6425,  -6474,  -6425,  -5935,  -4710,      1;
  1, -13334, -18038, -19263, -19704, -19704, -19263, -18038, -13334,      1;
  1, -46311, -59639, -64343, -65519, -65911, -65519, -64343, -59639, -46311, 1;
		

Links

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

Programs

Magma

A015442:= func< n | &+[7^j*Binomial(n-j,j): j in [0..Floor(n/2)]] >;
[A015442(k) - A015442(n) + A015442(n-k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 24 2019

Mathematica

(* Set of sequences q=0..10. This sequence is q=7. *)
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 *)
A015442[n_]:= Sum[7^j*Binomial[n-j, j], {j,0,(n+1)/2}]; T[n_, k_]:= T[n, k]= A015442[k] +A015442[n-k] -A015442[n]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 24 2019 *)

PARI

A015442(n) = sum(j=0,(n+1)\2, 7^j*binomial(n-j,j));
T(n,k) = A015442(k) - A015442(n) + A015442(n-k); \\ G. C. Greubel, Nov 24 2019

Sage

def A015442(n): return sum(7^j*binomial(n-j,j) for j in (0..floor(n/2)))
[[A015442(k) - A015442(n) + A015442(n-k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 24 2019

Formula

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