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

0,5

Links

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

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

Example

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;

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A099575 A173740 A028275 * A147289 A147566 A204621

Adjacent sequences: A173115 A173116 A173117 * A173119 A173120 A173121

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula, Feb 10 2010

Extensions

Edited by G. C. Greubel, Apr 27 2021

Status

approved