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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.
3
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))).
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
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