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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 (list; table; graph; refs; listen; history; text; internal format)
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: A019253 A019173 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

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Last modified June 15 00:00 EDT 2021. Contains 345041 sequences. (Running on oeis4.)