A172345
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, 7, 1, 1, 50, 50, 1, 1, 357, 2550, 357, 1, 1, 2549, 129999, 129999, 2549, 1, 1, 18200, 6627400, 47319636, 6627400, 18200, 1, 1, 129949, 337867400, 17224480052, 17224480052, 337867400, 129949, 1, 1, 927843, 17224610001, 6269758040364
Offset: 0
1;
1, 1;
1, 7, 1;
1, 50, 50, 1;
1, 357, 2550, 357, 1;
1, 2549, 129999, 129999, 2549, 1;
1, 18200, 6627400, 47319636, 6627400, 18200, 1;
1, 129949, 337867400, 17224480052, 17224480052, 337867400, 129949, 1;
1, 927843, 17224610001, 6269758040364, 44766423655148, 6269758040364, 17224610001, 927843, 1;
-
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
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;
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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}]
A172346
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=8.
Original entry on oeis.org
1, 1, 1, 1, 8, 1, 1, 65, 65, 1, 1, 528, 4290, 528, 1, 1, 4289, 283074, 283074, 4289, 1, 1, 34840, 18678595, 151727664, 18678595, 34840, 1, 1, 283009, 1232504195, 81326315267, 81326315267, 1232504195, 283009, 1, 1, 2298912, 81326598276
Offset: 0
1;
1, 1;
1, 8, 1;
1, 65, 65, 1;
1, 528, 4290, 528, 1;
1, 4289, 283074, 283074, 4289, 1;
1, 34840, 18678595, 151727664, 18678595, 34840, 1;
1, 283009, 1232504195, 81326315267, 81326315267, 1232504195, 283009, 1;
1, 2298912, 81326598276, 43591056675936, 354094776672518, 43591056675936, 81326598276, 2298912, 1;
-
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}]
A383719
a(n) = Pell(n) * Pell(n-1) * Pell(n-2) * Pell(n-3) * Pell(n-4) / 3480.
Original entry on oeis.org
1, 70, 5915, 482664, 39618670, 3248730940, 266442347522, 21851425660680, 1792084691254935, 146972777186757522, 12053560080255418725, 988538895611708641200, 81072243052956528402380, 6648912468496274313591800, 545291894670184984544154100, 44720584275276797753993516592
Offset: 5
-
pell(n) = ([2, 1; 1, 0]^n)[2, 1];
p(n, k) = prod(j=0, k-1, pell(n-j));
a(n) = p(n, 5)/p(5, 5);
-
def a(n): return ((1+sqrt(2))^(5*(n-5))*q_binomial(n, 5, -(3-2*sqrt(2)))).simplify_full()
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