A176261 Triangle T(n,k) = A006130(k) - A006130(n) + A006130(n-k) read by rows.

1, 1, 1, 1, -2, 1, 1, -2, -2, 1, 1, -11, -11, -11, 1, 1, -20, -29, -29, -20, 1, 1, -56, -74, -83, -74, -56, 1, 1, -119, -173, -191, -191, -173, -119, 1, 1, -290, -407, -461, -470, -461, -407, -290, 1, 1, -650, -938, -1055, -1100, -1100, -1055, -938, -650, 1

Offset: 0

Author

Roger L. Bagula, Apr 13 2010

Keywords

• sign
• tabl
• easy

Comments

Row sums are s(n) = {1, 2, 0, -2, -31, -96, -341, -964, -2784, -7484, -20041, ...}, obey s(n) = 3*s(n-1) + 3*s(n-2) - 11*s(n-3) - 3*s(n-4) + 9*s(n-5) and have g.f. (1-x+3*x^3-9*x^2)/((1-x)*(1-x-3*x^2)^2).

Examples

			Triangle begins as:
  1;
  1,     1;
  1,    -2,     1;
  1,    -2,    -2,     1;
  1,   -11,   -11,   -11,     1;
  1,   -20,   -29,   -29,   -20,     1;
  1,   -56,   -74,   -83,   -74,   -56,     1;
  1,  -119,  -173,  -191,  -191,  -173,  -119,     1;
  1,  -290,  -407,  -461,  -470,  -461,  -407,  -290,     1;
  1,  -650,  -938, -1055, -1100, -1100, -1055,  -938,  -650,     1;
  1, -1523, -2171, -2459, -2567, -2603, -2567, -2459, -2171, -1523, 1;
		

Links

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

Crossrefs

Cf. A006130.

Programs

Magma

A006130:= func< n | &+[Binomial(n-j,j)*3^j: j in [0..n]] >;
[A006130(k) -A006130(n) +A006130(n-k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 24 2019

Maple

A176261 := proc(n,k)
        A006130(k)-A006130(n)+A006130(n-k) ;
end proc; # R. J. Mathar, May 03 2013

Mathematica

A006130[n_]:= Sum[Binomial[n-j,j]*3^j, {j,0,n}]; T[n_,k_]:= A006130[k] - A006130[n] + A006130[n-k]; Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Nov 24 2019 *)

PARI

A006130(n) = sum(j=0,n,binomial(n-j,j)*3^j);
T(n,k) = A006130(k) -A006130(n) +A006130(n-k); \\ G. C. Greubel, Nov 24 2019

Sage

def A006130(n): return sum(binomial(n-j,j)*3^j for j in (0..n))
[[A006130(k) -A006130(n) +A006130(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).

T(n,k) = A006130(k) - A006130(n) + A006130(n-k), where A006130(n) = Sum_{j=0..n} binomial(n-j, j)*3^j. - G. C. Greubel, Nov 24 2019