A172353 -id:A172353 - cp's OEIS frontend

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.

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

Offset: 0

Roger L. Bagula, Feb 01 2010

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)) ).

			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;

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

Cf. A172353 (q=1), this sequence (q=2), A172359 (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, 2], {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,2) 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 = 2. - G. C. Greubel, May 09 2021

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

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.

Original entry on oeis.org

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

A172355 Triangle t(n,k) read by rows: generalized Padovan factorial ratios c(n)/(c(k)*c(n-k)) where c are partial products of a generalized Padovan sequence with multiplier m=5.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 6, 30, 6, 1, 1, 26, 156, 156, 26, 1, 1, 35, 910, 1092, 910, 35, 1, 1, 136, 4760, 24752, 24752, 4760, 136, 1, 1, 201, 27336, 191352, 829192, 191352, 27336, 201, 1, 1, 715, 143715, 3909048, 22802780, 22802780, 3909048, 143715, 715

Offset: 0

Roger L. Bagula, Feb 01 2010

Start from the generalized Padovan sequence f(n) = 0, 1, 1, 5, 6, 26, 35, 136, 201, 715, 1141, 3776,.. , f(n) = 5*f(n-2)+f(n-3), and its partial products c(n) = 1, 1, 1, 5, 30, 780, 27300, 3712800, 746272800, 533585052000.. Then t(n,k) = c(n)/(c(k)*c(n-k)).
Row sums are 1, 2, 3, 12, 44, 366, 2984, 59298, 1266972, 53712518, 2554657926,....
Note that rows n>= 14 contain fractions. - R. J. Mathar, Jul 05 2012

			1;
1, 1;
1, 1, 1;
1, 5, 5, 1;
1, 6, 30, 6, 1;
1, 26, 156, 156, 26, 1;
1, 35, 910, 1092, 910, 35, 1;
1, 136, 4760, 24752, 24752, 4760, 136, 1;
1, 201, 27336, 191352, 829192, 191352, 27336, 201, 1;
1, 715, 143715, 3909048, 22802780, 22802780, 3909048, 143715, 715, 1;
1, 1141, 815815, 32795763, 743370628, 1000691230, 743370628, 32795763, 815815, 1141, 1;

Cf. A010048, A172353

Clear[f, c, a, t];
f[0, a_] := 0; f[1, a_] := 1; f[2, a_] := 1;
f[n_, a_] := f[n, a] = a*f[n - 2, a] + f[n - 3, a];
c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
Table[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}], {a, 1, 10}];
Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]

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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.

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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.

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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.

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A172355 Triangle t(n,k) read by rows: generalized Padovan factorial ratios c(n)/(c(k)*c(n-k)) where c are partial products of a generalized Padovan sequence with multiplier m=5.

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