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A097939
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Sum of the smallest parts of all compositions of n.
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9
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1, 3, 6, 12, 22, 42, 79, 151, 291, 566, 1106, 2175, 4293, 8499, 16864, 33523, 66727, 132958, 265137, 529050, 1056169, 2109282, 4213710, 8419697, 16827079, 33634489, 67237513, 134424624, 268768414, 537407062, 1074605619, 2148875961, 4297212424, 8593556211, 17185713097, 34369170909
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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G.f.: Sum_{k>=1} x^k/(1-x-x^k).
a(n) = Sum_{r=0..n-1} Sum_{k=0..floor((n-r-1)/(r+1))} binomial(n-r(k+1)-1, k). - Paul Barry, Oct 08 2004
G.f.: (1-x)^2 * Sum_{k>=1} k*x^k/((x^k+x-1)*(x^(k+1)+x-1)). - Vladeta Jovovic, Apr 23 2006
G.f.: Sum_{n>=1} a*x^n/(1-a*x^n) (generalized Lambert series) where a=1/(1-x). - Joerg Arndt, Jan 30 2011
G.f.: Sum_{n>=1} (a*x)^n/(1-x^n) where a=1/(1-x). - Joerg Arndt, Jan 01 2013
G.f.: Sum_{n>=1} x^n * Sum_{d|n} 1/(1-x)^d. - Paul D. Hanna, Jul 18 2013
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MAPLE
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# second Maple Program:
b:= proc(n, m) option remember; `if`(n=0, m,
add(b(n-j, min(j, m)), j=1..n))
end:
a:= n-> b(n$2):
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MATHEMATICA
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Drop[ CoefficientList[ Series[ Sum[x^k/(1 - x - x^k), {k, 50}], {x, 0, 35}], x], 1] (* Robert G. Wilson v, Sep 08 2004 *)
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PROG
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(PARI)
N=66; x='x+O('x^N);
gf= sum(k=1, N, x^k/(1-x-x^k) );
Vec(gf)
(PARI) {a(n)=polcoeff(sum(m=1, n, x^m*sumdiv(m, d, 1/(1-x +x*O(x^n))^d) ), n)}
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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