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A255170
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a(n) = A087803(n) - n + 1.
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5
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1, 1, 2, 5, 13, 32, 79, 193, 478, 1196, 3037, 7802, 20287, 53259, 141069, 376449, 1011295, 2732453, 7421128, 20247355, 55469186, 152524366, 420807220, 1164532203, 3231706847, 8991343356, 25075077684, 70082143952, 196268698259, 550695545855, 1547867058852
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OFFSET
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1,3
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COMMENTS
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Conjectured extension of A199812: number of distinct values taken by w^w^...^w (with n w's and parentheses inserted in all possible ways) where w is the first transfinite ordinal omega. So far all known terms of A199812 (that is, 20 of them) coincide with this sequence. It is conjectured that A199812 is actually identical to this sequence, but it remains unproved, and is computationally difficult to check for n > 20.
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LINKS
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FORMULA
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a(n) = 1 - n + Sum_{k=1..n} A000081(k).
Recurrence: a(1) = 1, a(n+1) = a(n) + A000081(n+1) - 1.
Recurrence: a(1) = a(2) = 1, a(n) = A174145(n-1) + 2*a(n-1) - a(n-2).
Asymptotics: a(n) ~ c * d^n / n^(3/2), where c = A187770 / (1 - 1 / A051491) = 0.664861... and d = A051491 = 2.955765...
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EXAMPLE
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a(4) = 1 - 4 + Sum_{k=1..4} A000081(k) = 1 - 4 + 1 + 1 + 2 + 4 = 5.
a(5) = 1 - 5 + Sum_{k=1..5} A000081(k) = 1 - 5 + 1 + 1 + 2 + 4 + 9 = 13.
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MAPLE
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with(numtheory):
t:= proc(n) option remember; `if`(n<2, n, (add(add(
d*t(d), d=divisors(j))*t(n-j), j=1..n-1))/(n-1))
end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<2, 0,
add(b(n-i*j, i-1)*binomial(t(i)+j-1, j), j=0..n/i)))
end:
a:= proc(n) option remember; `if`(n<3, 1,
b(n-1$2) +2*a(n-1) -a(n-2))
end:
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MATHEMATICA
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t[1] = a[1] = 1; t[n_] := t[n] = Sum[k t[k] t[n - k m]/(n-1), {k, n}, {m, (n-1)/k}]; a[n_] := a[n] = a[n-1] + t[n] - 1; Table[a[n], {n, 40}] (* Vladimir Reshetnikov, Aug 12 2016 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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