A173117 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 = 1, read by rows.

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 5, 8, 5, 1, 1, 6, 14, 14, 6, 1, 1, 7, 20, 28, 20, 7, 1, 1, 8, 27, 49, 49, 27, 8, 1, 1, 9, 35, 76, 98, 76, 35, 9, 1, 1, 10, 44, 111, 175, 175, 111, 44, 10, 1, 1, 11, 54, 155, 286, 350, 286, 155, 54, 11, 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 = 1.

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 = 1. - G. C. Greubel, Apr 27 2021

Example

Triangle begins as:
  1;
  1,  1;
  1,  3,  1;
  1,  4,  4,   1;
  1,  5,  8,   5,   1;
  1,  6, 14,  14,   6,   1;
  1,  7, 20,  28,  20,   7,   1;
  1,  8, 27,  49,  49,  27,   8,   1;
  1,  9, 35,  76,  98,  76,  35,   9,  1;
  1, 10, 44, 111, 175, 175, 111,  44, 10,  1;
  1, 11, 54, 155, 286, 350, 286, 155, 54, 11, 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, 1], {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, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 27 2021

Crossrefs

Cf. A007318 (q=0), A072405 (q= -1), this sequence (q=1), A173118 (q=2), A173119 (q=3), A173120 (q= -4), A173122.

Sequence in context: A362036 A132735 A028262 * A050177 A013580 A147290

Adjacent sequences: A173114 A173115 A173116 * A173118 A173119 A173120

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula, Feb 10 2010

Extensions

Edited by G. C. Greubel, Apr 27 2021

Status

approved