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Search: id:A141540
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| A141540 |
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A triangular sequence of coefficients in a renormalized fractional factorial recursion ( a neo -combinatorial process):. |
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+0
1
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| 1, 1, 1, 1, 4, 1, 1, 6, 6, 1, 1, 8, 12, 8, 1, 1, 10, 20, 20, 10, 1, 1, 12, 30, 40, 30, 12, 1, 1, 14, 42, 70, 70, 42, 14, 1, 1, 16, 56, 112, 140, 112, 56, 16, 1, 1, 18, 72, 168, 252, 252, 168, 72, 18, 1, 1, 20, 90, 240, 420, 504, 420, 240, 90, 20, 1
(list; table; graph; listen)
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OFFSET
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1,5
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COMMENT
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Row sums are:
{1, 2, 6, 14, 30, 62, 126, 254, 510, 1022, 2046}.
This renormalization process is modeled of frequency scales in music
and renormalization processes in measured physical quantities
like mass and energy: specifically the mass like n->{0,23]:
m[n]=hbar*Gamma[n+1/2]*r[n]/r[n-1].
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FORMULA
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a(n) = A000045[n]; Renormalized factorial; f(n) = a(n)*n*f(n - 1)/a(n - 1); Neo-combination: t(n, m) = f(n)/(f(n - m)*f(m))
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EXAMPLE
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{1},
{1, 1},
{1, 4, 1},
{1, 6, 6, 1},
{1, 8, 12, 8, 1},
{1, 10, 20, 20, 10, 1},
{1, 12, 30, 40, 30, 12, 1},
{1, 14, 42, 70, 70, 42, 14, 1},
{1, 16, 56, 112, 140, 112, 56, 16, 1},
{1, 18, 72, 168, 252, 252, 168, 72, 18, 1},
{1, 20, 90, 240, 420, 504, 420, 240, 90, 20, 1}
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MATHEMATICA
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Clear[a, n, f, g] a[0] = 0; a[1] = 1; a[n_] := a[n] = a[n - 1] + a[n - 1] Table[a[n], {n, 0, 30}] (* renormalized fractional factorial recursion*) f[0] = 1; f[1] = 1; f[n_] := f[n] = a[n]*n*f[n - 1]/a[n - 1]; Table[f[n], {n, 0, 10}]; g[n_, m_] := g[n, m] = f[n]/(f[n - m]*f[m]); Table[Table[g[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]
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CROSSREFS
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Cf. A000045.
Sequence in context: A140262 A049702 A132046 this_sequence A143188 A102413 A144480
Adjacent sequences: A141537 A141538 A141539 this_sequence A141541 A141542 A141543
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KEYWORD
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nonn,uned,tabl
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AUTHOR
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Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Aug 15 2008
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