A172351 -id:A172351 - cp's OEIS frontend

A172350 Triangle t(n,k) read by rows: fibonomial ratios c(n)/(c(k)*c(n-k)) where c are partial products of a generalized Fibonacci sequence with multiplier m=5.

1, 1, 1, 1, 1, 1, 1, 6, 6, 1, 1, 11, 66, 11, 1, 1, 41, 451, 451, 41, 1, 1, 96, 3936, 7216, 3936, 96, 1, 1, 301, 28896, 197456, 197456, 28896, 301, 1, 1, 781, 235081, 3761296, 14019376, 3761296, 235081, 781, 1, 1, 2286, 1785366, 89565861, 781665696

Offset: 0

Roger L. Bagula and Gary W. Adamson, Feb 01 2010

Start from the generalized Fibonacci sequence A015440 and its partial products c(n) = 1, 1, 1, 6, 66, 2706, 259776, 78192576, 61068401856, 139602366642816... Then t(n,k) = c(n)/(c(k)*c(n-k)).

Row sums are 1, 2, 3, 14, 90, 986, 15282, 453308, 22013694, 1746038420, 222562828116,...

			1;
1, 1;
1, 1, 1;
1, 6, 6, 1;
1, 11, 66, 11, 1;
1, 41, 451, 451, 41, 1;
1, 96, 3936, 7216, 3936, 96, 1;
1, 301, 28896, 197456, 197456, 28896, 301, 1;
1, 781, 235081, 3761296, 14019376, 3761296, 235081, 781, 1;
1, 2286, 1785366, 89565861, 781665696, 781665696, 89565861, 1785366, 2286, 1;
1, 6191, 14152626, 1842200151, 50409295041, 118031520096, 50409295041, 1842200151, 14152626, 6191, 1;

Cf. A010048 (m=1), A015109 (m=2), A172349 (m=4), A172351 (m=6).

Clear[f, c, a, t];
f[0, a_] := 0; f[1, a_] := 1;
f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, 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}]

A172352 Triangle t(n,k) read by rows: fibonomial ratios c(n)/(c(k)*c(n-k)) where c are partial products of a generalized Fibonacci sequence with multiplier m=7.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 8, 1, 1, 15, 120, 15, 1, 1, 71, 1065, 1065, 71, 1, 1, 176, 12496, 23430, 12496, 176, 1, 1, 673, 118448, 1051226, 1051226, 118448, 673, 1, 1, 1905, 1282065, 28205430, 133505702, 28205430, 1282065, 1905, 1, 1, 6616, 12603480

Offset: 0

Roger L. Bagula and Gary W. Adamson, Feb 01 2010

Start from the generalized Fibonacci sequence A015442 and its partial products c(n) = 1, 1, 1, 8, 120, 8520, 1499520, 1009176960, 1922482108800... Then t(n,k) = c(n)/(c(k)*c(n-k)).

Row sums are 1, 2, 3, 18, 152, 2274, 48776, 2340696, 192484504, 27026705688, 6379354108992,...

			1;
1, 1;
1, 1, 1;
1, 8, 8, 1;
1, 15, 120, 15, 1;
1, 71, 1065, 1065, 71, 1;
1, 176, 12496, 23430, 12496, 176, 1;
1, 673, 118448, 1051226, 1051226, 118448, 673, 1;
1, 1905, 1282065, 28205430, 133505702, 28205430, 1282065, 1905, 1;
1, 6616, 12603480, 1060267755, 12440474992, 12440474992, 1060267755, 12603480, 6616, 1;

CF. A010048 (m=1), A015109 (m=2), A172351 (m=6).

Clear[f, c, a, t];
f[0, a_] := 0; f[1, a_] := 1;
f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, 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}]

cp's OEIS Frontend

A172350 Triangle t(n,k) read by rows: fibonomial ratios c(n)/(c(k)*c(n-k)) where c are partial products of a generalized Fibonacci sequence with multiplier m=5.

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