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A172358 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, 3, 3, 3, 1, 1, 3, 9, 9, 3, 1, 1, 5, 15, 45, 15, 5, 1, 1, 9, 45, 135, 135, 45, 9, 1, 1, 11, 99, 495, 495, 495, 99, 11, 1, 1, 19, 209, 1881, 3135, 3135, 1881, 209, 19, 1, 1, 29, 551, 6061, 18183, 30305, 18183, 6061, 551, 29, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,12

COMMENTS

Start from the sequence A159284 and its partial products c(n) = 1, 1, 1, 1, 3, 9, 45, 405, 4455, 84645, 2454705, ... . 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 = 2. - G. C. Greubel, May 09 2021

EXAMPLE

Triangle begins as:

  1;

  1,  1;

  1,  1,   1;

  1,  1,   1,    1;

  1,  3,   3,    3,     1;

  1,  3,   9,    9,     3,     1;

  1,  5,  15,   45,    15,     5,     1;

  1,  9,  45,  135,   135,    45,     9,    1;

  1, 11,  99,  495,   495,   495,    99,   11,   1;

  1, 19, 209, 1881,  3135,  3135,  1881,  209,  19,  1;

  1, 29, 551, 6061, 18183, 30305, 18183, 6061, 551, 29, 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, 2], {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, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 09 2021

CROSSREFS

Cf. A172353 (q=1), this sequence (q=2), A172359 (q=4), A172360 (q=5).

Sequence in context: A190906 A080311 A135368 * A119560 A172364 A323596

Adjacent sequences:  A172355 A172356 A172357 * A172359 A172360 A172361

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 19 09:23 EDT 2021. Contains 345126 sequences. (Running on oeis4.)