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A000713
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EULER transform of 3, 2, 2, 2, 2, 2, 2, 2, ...
(Formerly M2731 N1096)
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7
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1, 3, 8, 18, 38, 74, 139, 249, 434, 734, 1215, 1967, 3132, 4902, 7567, 11523, 17345, 25815, 38045, 55535, 80377, 115379, 164389, 232539, 326774, 456286, 633373, 874213, 1200228, 1639418, 2228546, 3015360, 4062065, 5448995, 7280060, 9688718, 12846507, 16972577
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OFFSET
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0,2
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COMMENTS
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Equals the number of partitions of n with 1's of three kinds and all parts >1 of two kinds. - Gregory L. Simay, Mar 25 2018
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REFERENCES
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H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 122.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*Pi*3^(1/4)*n^(3/4)).
(End)
G.f.: exp(Sum_{k>=1} (2*sigma_1(k) + 1)*x^k/k). - Ilya Gutkovskiy, Aug 21 2018
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MAPLE
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with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> `if`(n<2, 3, 2)): seq(a(n), n=0..40); # Alois P. Heinz, Sep 08 2008
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MATHEMATICA
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nn=20; g=Product[1/(1-x^i), {i, 1, nn}]; c=1/(1-x); CoefficientList[Series[g^2/(1-x), {x, 0, nn}], x] (* Geoffrey Critzer, Apr 19 2012 *)
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PROG
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(PARI) x='x+O('x^66); Vec(1/((1-x)*eta(x)^2)) \\ Joerg Arndt, May 01 2013
(Python)
from functools import lru_cache
from sympy import divisor_sigma
@lru_cache(maxsize=None)
def A000713(n): return sum(A000713(k)*((divisor_sigma(n-k)<<1)+1) for k in range(n))//n if n else 1 # Chai Wah Wu, Sep 25 2023
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CROSSREFS
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KEYWORD
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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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