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Search: id:A135686
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| A135686 |
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Two sequence type recursion of a generalized Stirling number type using the Fibonacci sequence: general form is like Hermite recursions. |
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+0
1
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| 1, 1, 3, 6, 21, 69, 342, 1791, 13419, 111924, 1306215, 17423271, 321771366, 6890344533, 203170877793, 7003940931864, 331467832767285, 18429651200703861, 1404296660000722446, 126080887032762342111
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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This sequence is related to the Hermite-Bessel recursion generalizations and to the Stirling numbers.
I noticed that:
a(n)=2*(a(n - 1) - (n - 1)*a(n - 2)
and
a[n] = (n + 1)*a[n - 1] + a[n - 2]
were similar.
What I discovered was that:
a(n)=k*a(n-1)+f[n]*a[n-2]
was Hermite like in special cases of k and f[n]
and
a(n)=f[n]*a(n-1)+k*a[n-2]
Bessel for special cases of k and f[n].
If you generalize this to f[n] equaling a second recursive sequence b(n)
and k=1,
you get this result.
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FORMULA
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b(n)=A000045(n); a(n) = a(n-1)+A000045(n)*a(n-1);
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MATHEMATICA
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b[0] = 1; b[1] = 1; b[n_] := b[n] = b[n - 1] + b[n - 2]; a[0] = 1; a[1] = 1; a[n_] := a[n] = a[n - 1] + b[n]*a[n - 2]; Table[a[n], {n, 0, 30}]
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CROSSREFS
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Cf. A000045, A135829.
Adjacent sequences: A135683 A135684 A135685 this_sequence A135687 A135688 A135689
Sequence in context: A112520 A054878 A084567 this_sequence A025229 A073951 A032098
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KEYWORD
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nonn,uned
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 18 2008
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