A173118 Triangle T(n, k, q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with T(n,0) = T(n,n) = 1 for q = 2, read by rows.

1, 1, 1, 1, 4, 1, 1, 5, 5, 1, 1, 6, 10, 6, 1, 1, 7, 20, 20, 7, 1, 1, 8, 27, 40, 27, 8, 1, 1, 9, 35, 75, 75, 35, 9, 1, 1, 10, 44, 110, 150, 110, 44, 10, 1, 1, 11, 54, 154, 276, 276, 154, 54, 11, 1, 1, 12, 65, 208, 430, 552, 430, 208, 65, 12, 1

Offset: 0

Author

Roger L. Bagula, Feb 10 2010

Keywords

• nonn
• tabl
• easy
• less

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  4,  1;
  1,  5,  5,   1;
  1,  6, 10,   6,   1;
  1,  7, 20,  20,   7,   1;
  1,  8, 27,  40,  27,   8,   1;
  1,  9, 35,  75,  75,  35,   9,   1;
  1, 10, 44, 110, 150, 110,  44,  10,  1;
  1, 11, 54, 154, 276, 276, 154,  54, 11,  1;
  1, 12, 65, 208, 430, 552, 430, 208, 65, 12, 1;
		

Links

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

Crossrefs

Cf. A007318 (q=0), A072405 (q= -1), A173117 (q=1), this sequence (q=2), A173119 (q=3), A173120 (q= -4), A173122.

Programs

Mathematica

T[n_, k_, q_]:= If[k==0 || k==n, 1, q*Boole[n==2] + Sum[q^j*Binomial[n-2*j, k-j]*Boole[n>2*j], {j,0,5}]];
Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Apr 27 2021 *)

Sage

@CachedFunction
def T(n,k,q): return 1 if (k==0 or k==n) else q*bool(n==2) + sum( q^j*binomial(n-2*j, k-j)*bool(n>2*j) for j in (0..5) )
flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 27 2021

Formula

T(n, k, q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with T(n,0) = T(n,n) = 1 for q = 2.

Sum_{k=0..n} T(n, k, q) = [n=0] + q*[n=2] + Sum_{j=0..5} q^j*2^(n-2*j)*[n > 2*j] for q = 2. - G. C. Greubel, Apr 27 2021

Extensions

Edited by G. C. Greubel, Apr 27 2021