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A014153
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G.f.: 1/[(1-x)^2*product((1-x^k),k=1..infinity)].
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13
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1, 3, 7, 14, 26, 45, 75, 120, 187, 284, 423, 618, 890, 1263, 1771, 2455, 3370, 4582, 6179, 8266, 10980, 14486, 18994, 24757, 32095, 41391, 53123, 67865, 86325, 109350, 137979, 173450, 217270, 271233, 337506, 418662, 517795, 638565, 785350, 963320, 1178628
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OFFSET
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0,2
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COMMENTS
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Number of partitions of n with three kinds of 1. E.g. a(2)=7 because we have 2, 1+1, 1+1', 1+1", 1'+1', 1'+1", 1"+1". - Emeric Deutsch, Mar 22 2005
Partial sums of the partial sums of the partition numbers A000041. Partial sums of A000070. Euler transform of 3,1,1,1,...
Also sum of parts, counted without multiplicity, in all partitions of n, offset 1. Also Sum phi(p), where the sum is taken over all parts p of all partitions of n, offset 1. - Vladeta Jovovic, Mar 26 2005
Equals row sums of triangle A141157. - Gary W. Adamson, Jun 12 2008
A014153 convolved with A010815 = (1, 2, 3,...). n-th partial sum sequence of A000041 convolved with A010815 = (n-1)-th column of Pascal's triangle, starting (1, n,...). [From Gary W. Adamson, Nov 09 2008]
a(n) is also the sum of all parts of the (n+1)st column of a version of the shell model of partitions in which each shell has its parts aligned to the right margin (Cf. A210953, A210970, A135010). Also rows of triangle A210952 converge to this sequence. - Omar E. Pol, May 25 2012
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..1000
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MAPLE
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with(numtheory):
a:= proc(n) option remember;
`if`(n=0, 1, add((2+sigma(j)) *a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..40); # Alois P. Heinz, Feb 13 2012
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CROSSREFS
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Cf. A000041, A000070, A141157.
Cf. A010815. - Gary W. Adamson, Nov 09 2008
Sequence in context: A207381 A008646 A036830 * A001924 A079921 A014168
Adjacent sequences: A014150 A014151 A014152 * A014154 A014155 A014156
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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