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A172360 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, 6, 6, 6, 1, 1, 6, 36, 36, 6, 1, 1, 11, 66, 396, 66, 11, 1, 1, 36, 396, 2376, 2376, 396, 36, 1, 1, 41, 1476, 16236, 16236, 16236, 1476, 41, 1, 1, 91, 3731, 134316, 246246, 246246, 134316, 3731, 91, 1, 1, 221, 20111, 824551, 4947306, 9070061, 4947306, 824551, 20111, 221, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,12

COMMENTS

Start from the sequence 0, 1, 1, 1, 6, 6, 11, 36, 41, 91, 221, 296, 676, 1401, 2156, ..., f(n) = f(n-2) + 5*f(n-3), and its partial products c(n) = 1, 1, 1, 1, 6, 36, 396, 14256, 584496, 53189136, ... . 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 = 5. - G. C. Greubel, May 09 2021

EXAMPLE

Triangle begins as:

  1;

  1,   1;

  1,   1,     1;

  1,   1,     1,      1;

  1,   6,     6,      6,       1;

  1,   6,    36,     36,       6,       1;

  1,  11,    66,    396,      66,      11,       1;

  1,  36,   396,   2376,    2376,     396,      36,      1;

  1,  41,  1476,  16236,   16236,   16236,    1476,     41,     1;

  1,  91,  3731, 134316,  246246,  246246,  134316,   3731,    91,   1;

  1, 221, 20111, 824551, 4947306, 9070061, 4947306, 824551, 20111, 221, 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, 5], {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, 5) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 09 2021

CROSSREFS

Cf. A172353 (q=1), A172358 (q=2), A172359 (q=4), this sequence (q=5).

Sequence in context: A019180 A019103 A272619 * A175288 A153509 A248093

Adjacent sequences:  A172357 A172358 A172359 * A172361 A172362 A172363

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.)