A172358 -id:A172358 - cp's OEIS frontend

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.

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

Roger L. Bagula, Feb 01 2010

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))).

			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;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A172353 (q=1), A172358 (q=2), this sequence (q=4), A172360 (q=5).

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 *)

@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

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

Definition corrected to give integral terms by G. C. Greubel, May 09 2021

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.

Original entry on oeis.org

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

Offset: 0

Roger L. Bagula, Feb 01 2010

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))).

			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;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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

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 *)

@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

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

Definition corrected to give integral terms by G. C. Greubel, May 09 2021

cp's OEIS Frontend

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.

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