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A174411
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A product triangle sequence based on:a=1;f(n, a) = f(n - 1, a) + a*f(n - 2, a); c(n,a)=If[n == 0, 1, Product[f(i, a)*f(i + 1, a), {i, 1, n}]]; t(n,m,a)=If[Floor[n/2] >= m, c(n, a)/c(n - m, a), c(n, a)/c(m, a)]
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0
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1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 15, 90, 15, 1, 1, 40, 600, 600, 40, 1, 1, 104, 4160, 62400, 4160, 104, 1, 1, 273, 28392, 1135680, 1135680, 28392, 273, 1, 1, 714, 194922, 20271888, 810875520, 20271888, 194922, 714, 1, 1, 1870, 1335180, 364504140
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OFFSET
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0,5
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COMMENTS
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Row sums are:
{1, 2, 4, 14, 122, 1282, 70930, 2328692, 851810570, 76548543502,
189143352851092,...}
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LINKS
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FORMULA
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a=1;
f(n, a) = f(n - 1, a) + a*f(n - 2, a);
c(n,a)=If[n == 0, 1, Product[f(i, a)*f(i + 1, a), {i, 1, n}]];
t(n,m,a)=If[Floor[n/2] >= m, c(n, a)/c(n - m, a), c(n, a)/c(m, a)]
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EXAMPLE
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{1},
{1, 1},
{1, 2, 1},
{1, 6, 6, 1},
{1, 15, 90, 15, 1},
{1, 40, 600, 600, 40, 1},
{1, 104, 4160, 62400, 4160, 104, 1},
{1, 273, 28392, 1135680, 1135680, 28392, 273, 1},
{1, 714, 194922, 20271888, 810875520, 20271888, 194922, 714, 1},
{1, 1870, 1335180, 364504140, 37908430560, 37908430560, 364504140, 1335180, 1870, 1},
{1, 4895, 9153650, 6535706100, 1784247765300, 185561767591200, 1784247765300, 6535706100, 9153650, 4895, 1}
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MATHEMATICA
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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]*f[i + 1, a], {i, 1, n}]];
t[n_, m_, q_] = If[Floor[n/2] >= m, c[n, q]/c[n - m, q], c[n, q]/c[m, q]];
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 1, 10}]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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