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A305853
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Inverse Weigh transform of the Fubini numbers (ordered Bell numbers, A000670).
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3
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1, 3, 10, 62, 446, 3975, 41098, 484152, 6390488, 93419965, 1498268466, 26159940522, 494036061550, 10035451747919, 218207845446062, 5057251219752612, 124462048466812950, 3241773988594489244, 89093816361187396674, 2576652694087236419386, 78224564280680539732266
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graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 1..424
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FORMULA
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Product_{k>=1} (1+x^k)^a(k) = Sum_{n>=0} A000670(n) * x^n.
a(n) ~ n! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Sep 10 2019
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MAPLE
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g:= proc(n) option remember; `if`(n=0, 1,
add(g(n-j)*binomial(n, j), j=1..n))
end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= proc(n) option remember; g(n)-b(n, n-1) end:
seq(a(n), n=1..30);
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MATHEMATICA
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g[n_] := g[n] = If[n == 0, 1,
Sum[g[n - j] Binomial[n, j], {j, 1, n}]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
Sum[Binomial[a[i], j] b[n - i j, i - 1], {j, 0, n/i}]]];
a[n_] := a[n] = g[n] - b[n, n - 1];
a /@ Range[1, 30] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz *)
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CROSSREFS
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Cf. A000670, A095993, A305846, A305852.
Sequence in context: A034889 A228773 A260969 * A160921 A042705 A041014
Adjacent sequences: A305850 A305851 A305852 * A305854 A305855 A305856
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KEYWORD
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nonn
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AUTHOR
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Alois P. Heinz, Jun 11 2018
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STATUS
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approved
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