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A214948
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a(n) is the sum over all proper integer partitions of n of the previous terms.
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2
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1, 2, 6, 19, 51, 148, 395, 1095, 2945, 8020, 21597, 58518, 157746, 426250, 1149832, 3104236, 8375167, 22603530, 60988687, 164579663, 444082316, 1198312390, 3233419264, 8724918311, 23542640336, 63526028693, 171413973501, 462531951559, 1248062990751, 3367686427976
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OFFSET
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1,2
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COMMENTS
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By "proper integer partition", one means that the case {n} is excluded for having only one part, equal to the number partitioned.
The growth of this function is exponential a(n) -> c * exp(n). [This is not correct, a(n) ~ c * d^n, where d = A246828 = 2.69832910647421123126399... and c = 0.39308289517441096263558422597609193642795355676880812197435683468376... - Vaclav Kotesovec, Dec 27 2023]
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LINKS
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FORMULA
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a(n) = sum( sum( a(i), i in p) , p in P*(n)) where P*(n) is the set of all integer partitions of n excluding {n}, p is a partition of P*(n), i is a part of p.
a(n) ~ exp(k) * a(n-1), k = 0.992632731... (conjecture). - Bill McEachen, Dec 26 2023
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EXAMPLE
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a(4) = (a(3)+a(1))+(a(2)+a(2))+(a(2)+a(1)+a(1))+(a(1)+a(1)+a(1)+a(1)) = (6 + 1) + (2 + 2) + (2 + 2*1) + (4*1) = 7 + 4 + 4 + 4 = 19.
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MAPLE
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b:= proc(n, i) option remember; `if`(n<2, [1, n], `if`(i<1, 0,
b(n, i-1)+(p-> p+[0, p[1]*a(i)])(b(n-i, min(n-i, i)))))
end:
a:= n-> b(n, n-1)[2]:
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MATHEMATICA
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Clear[a]; a[1] := 1; a[n_Integer] :=
a[n] = Plus @@ Map[Function[p, Plus @@ Map[a, p]], Drop[IntegerPartitions[n], 1]]; Table[ a[n], {n, 1, 30}]
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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STATUS
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approved
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