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A000716
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Number of partitions of n into parts of 3 kinds.
(Formerly M2788 N1123)
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30
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1, 3, 9, 22, 51, 108, 221, 429, 810, 1479, 2640, 4599, 7868, 13209, 21843, 35581, 57222, 90882, 142769, 221910, 341649, 521196, 788460, 1183221, 1762462, 2606604, 3829437, 5590110, 8111346, 11701998, 16790136, 23964594, 34034391, 48104069, 67679109, 94800537, 132230021, 183686994, 254170332
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OFFSET
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0,2
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COMMENTS
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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.
Moreno, Carlos J., Partitions, congruences and Kac-Moody Lie algebras. Preprint, 37pp., no date. See Table I.
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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G.f.: Product_{m>=1} 1/(1-x^m)^3.
EULER transform of 3, 3, 3, 3, 3, 3, 3, 3, ...
a(0)=1, a(n) = 1/n*Sum_{k=0..n-1} 3*a(k)*sigma_1(n-k). - Joerg Arndt, Feb 05 2011
a(n) ~ exp(Pi * sqrt(2*n)) / (8 * sqrt(2) * n^(3/2)) * (1 - (3/Pi + Pi/8) / sqrt(2*n)). - Vaclav Kotesovec, Feb 28 2015, extended Jan 16 2017
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MAPLE
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with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(add(d*3, d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..40); # Alois P. Heinz, May 20 2013
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MATHEMATICA
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a[0] = 1; a[n_] := a[n] = 1/n*Sum[3*a[k]*DivisorSigma[1, n-k], {k, 0, n-1}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 03 2014, after Joerg Arndt *)
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PROG
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(PARI) Vec(1/eta('x+O('x^66))^3) \\ Joerg Arndt, Apr 28 2013
(Python)
from functools import lru_cache
from sympy import divisor_sigma
@lru_cache(maxsize=None)
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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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