A172359 Triangle read by rows: T(n,k) = round(c(n)/(c(k)*c(n-k))) where c are partial products of a sequence defined in comments.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 5, 5, 1, 1, 5, 25, 25, 5, 1, 1, 9, 45, 225, 45, 9, 1, 1, 25, 225, 1125, 1125, 225, 25, 1, 1, 29, 725, 6525, 6525, 6525, 725, 29, 1, 1, 61, 1769, 44225, 79605, 79605, 44225, 1769, 61, 1, 1, 129, 7869, 228201, 1141005, 2053809, 1141005, 228201, 7869, 129, 1

Offset

0,12

Comments

Start from the sequence 0, 1, 1, 1, 5, 5, 9, 25, 29, 61, 129, 177, 373, 693, 1081, 2185, 3853, ..., f(n) = f(n-2) + 4*f(n-3) and its partial products c(n) = 1, 1, 1, 1, 5, 25, 225, 5625, 163125, 9950625, ... . Then T(n,k) = round(c(n)/(c(k)*c(n-k))).

Links

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

Formula

T(n, k, q) = round(c(n,q)/(c(k,q)*c(n-k,q)), where c(n,q) = Product_{j=1..n} f(j,q), f(n, q) = f(n-2, q) + q*f(n-3, q), f(0,q)=0, f(1,q) = f(2,q) = 1, and q = 4. - G. C. Greubel, May 09 2021

Example

Triangle begins as:
  1;
  1,   1;
  1,   1,    1;
  1,   1,    1,      1;
  1,   5,    5,      5,       1;
  1,   5,   25,     25,       5,       1;
  1,   9,   45,    225,      45,       9,       1;
  1,  25,  225,   1125,    1125,     225,      25,      1;
  1,  29,  725,   6525,    6525,    6525,     725,     29,    1;
  1,  61, 1769,  44225,   79605,   79605,   44225,   1769,   61,   1;
  1, 129, 7869, 228201, 1141005, 2053809, 1141005, 228201, 7869, 129, 1;

Mathematica

f[n_, q_]:= f[n, q]= If[n<3, Fibonacci[n], f[n-2, q] + q*f[n-3, q]];
c[n_, q_]:= Product[f[j, q], {j, n}];
T[n_, k_, q_]:= Round[c[n, q]/(c[k, q]*c[n-k, q])];
Table[T[n, k, 4], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, May 09 2021 *)

Prog

(Sage)
@CachedFunction
def f(n, q): return fibonacci(n) if (n<3) else f(n-2, q) + q*f(n-3, q)
def c(n, q): return product( f(j, q) for j in (1..n) )
def T(n, k, q): return round(c(n, q)/(c(k, q)*c(n-k, q)))
flatten([[T(n, k, 4) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 09 2021

Crossrefs

Cf. A172353 (q=1), A172358 (q=2), this sequence (q=4), A172360 (q=5).

Sequence in context: A019173 A374753 A230192 * A093796 A021647 A181668

Adjacent sequences: A172356 A172357 A172358 * A172360 A172361 A172362

Keyword

nonn,tabl,less

Author

Roger L. Bagula, Feb 01 2010

Extensions

Definition corrected to give integral terms by G. C. Greubel, May 09 2021

Status

approved