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A172368 Triangle read by rows: T(n,k) = round(c(n)/(c(k)*c(n-k))) where c is a sequence defined in comments. 2
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 1, 1, 5, 15, 15, 15, 5, 1, 1, 7, 35, 105, 105, 35, 7, 1, 1, 9, 63, 315, 945, 315, 63, 9, 1, 1, 15, 135, 945, 4725, 4725, 945, 135, 15, 1, 1, 25, 375, 3375, 23625, 39375, 23625, 3375, 375, 25, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,17
COMMENTS
Start from A052942 and its partial products c(n) = 1, 1, 1, 1, 1, 3, 15, 105, 945, ... . Then T(n,k) = round(c(n)/(c(k)*c(n-k))).
LINKS
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-1, q) + q*f(n-4, q), f(0, q) = 0, f(1, q) = f(2, q) = f(3, q) = 1, and q = 2. - G. C. Greubel, May 08 2021
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 1, 1;
1, 1, 1, 1;
1, 1, 1, 1, 1;
1, 3, 3, 3, 3, 1;
1, 5, 15, 15, 15, 5, 1;
1, 7, 35, 105, 105, 35, 7, 1;
1, 9, 63, 315, 945, 315, 63, 9, 1;
1, 15, 135, 945, 4725, 4725, 945, 135, 15, 1;
1, 25, 375, 3375, 23625, 39375, 23625, 3375, 375, 25, 1;
MATHEMATICA
f[n_, q_]:= f[n, q]= If[n==0, 0, If[n<4, 1, f[n-1, q] + q*f[n-4, 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 08 2021 *)
PROG
(Sage)
@CachedFunction
def f(n, q): return 0 if (n==0) else 1 if (n<4) else f(n-1, q) + q*f(n-4, 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 08 2021
CROSSREFS
Cf. A052942, A172363 (q=1), this sequence (q=2), A172369 (q=3).
Sequence in context: A180560 A320085 A358691 * A138070 A081334 A106694
KEYWORD
nonn,tabl,less
AUTHOR
Roger L. Bagula, Feb 01 2010
EXTENSIONS
Definition corrected to give integral terms, G. C. Greubel, May 08 2021
STATUS
approved

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Last modified July 13 02:50 EDT 2024. Contains 374265 sequences. (Running on oeis4.)