A172342 -id:A172342 - cp's OEIS frontend

A172343 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=6.

1, 1, 1, 1, 6, 1, 1, 37, 37, 1, 1, 228, 1406, 228, 1, 1, 1405, 53390, 53390, 1405, 1, 1, 8658, 2027415, 12493260, 2027415, 8658, 1, 1, 53353, 76988379, 2923477635, 2923477635, 76988379, 53353, 1, 1, 328776, 2923530988, 684106251192

Offset: 0

Roger L. Bagula, Feb 01 2010

Start from the generalized Fibonacci sequence A005668 and its partial products c(n) = 1, 1, 6, 222, 50616, 71115480, 615717825840, 32850393162041520... Then t(n,k) = c(n)/(c(k)*c(n-k)).
Row sums are 1, 2, 8, 76, 1864, 109592, 16565408, 6001038736, 5589714971584,
12478331908166432, 71624411004755875328,...

			1;
1, 1;
1, 6, 1;
1, 37, 37, 1;
1, 228, 1406, 228, 1;
1, 1405, 53390, 53390, 1405, 1;
1, 8658, 2027415, 12493260, 2027415, 8658, 1;
1, 53353, 76988379, 2923477635, 2923477635, 76988379, 53353, 1;
1, 328776, 2923530988, 684106251192, 4215654749670, 684106251192, 2923530988, 328776, 1;

Cf. A010048 (m=1), A099927 (m=2), A172342 (m=5), A172345 (m=7).

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

A172339 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=3.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 10, 10, 1, 1, 33, 110, 33, 1, 1, 109, 1199, 1199, 109, 1, 1, 360, 13080, 43164, 13080, 360, 1, 1, 1189, 142680, 1555212, 1555212, 142680, 1189, 1, 1, 3927, 1556401, 56030436, 185070228, 56030436, 1556401, 3927, 1, 1, 12970, 16977730

Offset: 0

Roger L. Bagula, Feb 01 2010

Build with the recipe of A010048 (m=1) and A099927 (m=2).
Start from the generalized Fibonacci sequence A006190 and its partial products c(n) = 1, 1, 3, 30, 990, 107910, 38847600, 46189796400,... Then t(n,k) = c(n)/(c(k)*c(n-k)).
Row sums are 1, 2, 5, 22, 178, 2618, 70046, 3398164, 300251758, 48114604076,
14041125439724,...

			1;
1, 1;
1, 3, 1;
1, 10, 10, 1;
1, 33, 110, 33, 1;
1, 109, 1199, 1199, 109, 1;
1, 360, 13080, 43164, 13080, 360, 1;
1, 1189, 142680, 1555212, 1555212, 142680, 1189, 1;
1, 3927, 1556401, 56030436, 185070228, 56030436, 1556401, 3927, 1;
1, 12970, 16977730, 2018652097, 22021659240, 22021659240, 2018652097, 16977730, 12970, 1;

Cf. A010048 (m=1), A099927 (m=2), A034802 (m=4), A172342 (m=5).

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

A172343 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=6.

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