A173917 A double product sequence based on a=2; f(n,a) = f(n-1,a) + a*f(n-2,a).

1, 1, 1, 1, 3, 1, 1, 15, 15, 1, 1, 55, 275, 55, 1, 1, 231, 4235, 4235, 231, 1, 1, 903, 69531, 254947, 69531, 903, 1, 1, 3655, 1100155, 16942387, 16942387, 1100155, 3655, 1, 1, 14535, 17708475, 1066050195, 4477410819, 1066050195, 17708475, 14535, 1

Offset

0,5

Links

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

Formula

c(n,a) = 1 if n = 0, Product_{i=1..n} f(i, a)*f(i+1, a) otherwise.

T(n,k) = Product_{i=1..k} ((q^(n+1-i)-1) / (q^i-1)) * ((q^(n+2-i)-1) / (q^(i+1)-1)) for 0 <= k <= n with q = -2 and the empty product 1 (k=0). - Werner Schulte, Nov 14 2018

Example

Triangle begins:
  1;
  1,     1;
  1,     3,        1;
  1,    15,       15,          1;
  1,    55,      275,         55,          1;
  1,   231,     4235,       4235,        231,          1;
  1,   903,    69531,     254947,      69531,        903,        1;
  1,  3655,  1100155,   16942387,   16942387,    1100155,     3655,     1;
  1, 14535, 17708475, 1066050195, 4477410819, 1066050195, 17708475, 14535, 1;
   ...

Mathematica

f[0, a_] := 0; f[1, a_] := 1;
f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, a];
c[n_, a_] := If[n == 0, 1, Product[f[i, a]*f[i + 1, a], {i, 1, n}]];
w[n_, m_, q_] := c[n, q]/(c[m, q]*c[n - m, q]);
Table[Table[Table[w[n, m, q], {m, 0, n}], {n, 0, 10}], {q, 1, 12}];
Table[Flatten[Table[Table[w[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 1, 12}]
Table[Product[QBinomial[n+k, k+j, -2]/QBinomial[n+k-j, k, -2], {k, 0, 1}], {n, 0, 10}, {j, 0, n}]//Flatten (* G. C. Greubel, Nov 21 2018 *)

Prog

(PARI) T(n, k)={prod(i=0, k-1, (((-2)^(n-i)-1) / ((-2)^(i+1)-1) * ((-2)^(n+1-i)-1) / ((-2)^(i+2)-1)))} \\ Andrew Howroyd, Nov 12 2018
(Magma) q:=-2; [[k le 0 select 1 else (&*[((q^(n+1-i)-1)/(q^i-1))*((q^(n+2-i)-1)/(q^(i+1)-1)): i in [1..k]]) : k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 21 2018
(Sage) [[prod(q_binomial(n+k, k+j, -2)/q_binomial(n+k-j, k, -2) for k in (0..1)) for j in range(n+1)] for n in range(10)] # G. C. Greubel, Nov 21 2018

Crossrefs

Cf. A156916 (q=2).

Sequence in context: A370691 A326800 A176225 * A174410 A156690 A228900

Adjacent sequences: A173914 A173915 A173916 * A173918 A173919 A173920

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 02 2010

Status

approved