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A047967
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Number of partitions of n with some part repeated.
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13
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0, 0, 1, 1, 3, 4, 7, 10, 16, 22, 32, 44, 62, 83, 113, 149, 199, 259, 339, 436, 563, 716, 913, 1151, 1453, 1816, 2271, 2818, 3496, 4309, 5308, 6502, 7959, 9695, 11798, 14298, 17309, 20877, 25151, 30203, 36225, 43323, 51748, 61651, 73359
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Also number of partitions of n with at least one even part. - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 10 2003. Example: a(5)=4 because we have [4,1],[3,2],[2,2,1] and [2,1,1,1] ([5],[3,1,1] and [1,1,1,1,1] do not qualify). - Emeric Deutsch, Mar 30 2006
Also number of partitions of n (where it is assumed that the least part is 0) such that at least one difference is at least two. Example: a(5)=4 because we have [5,0],[4,1,0],[3,2,0] and [3,1,1,0] ([2,2,1,0],[2,1,1,1,0] and [1,1,1,1,1,0] do not qualify). - Emeric Deutsch, Mar 30 2006
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LINKS
| H. Bottomley, Illustration for A000009, A000041, A047967
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FORMULA
| G.f.: sum(x^(2*k)*product(1+x^j, j=k+1..infinity)/product(1-x^j, j=1..k), k=1..infinity) = sum(x^(2k)/(product(1-x^j, j=1..2*k)*product(1-x^(2*j+1), j=k..infinity) ), k=1..infinity). - Emeric Deutsch, Mar 30 2006
G.f.: 1/P(x) - P(x^2)/P(x) where P(x)=prod(k>=1, 1-x^k ). [Joerg Arndt, Jun 21 2011]
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EXAMPLE
| a(5) = 4 because we have [3,1,1], [2,2,1], [2,1,1,1] and [1,1,1,1,1] ([5], [4,1] and [3,2] do not qualify).
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MAPLE
| g:=sum(x^(2*k)*product(1+x^j, j=k+1..70)/product(1-x^j, j=1..k), k=1..40): gser:=series(g, x=0, 50): seq(coeff(gser, x, n), n=0..44); - Emeric Deutsch, Mar 30 2006
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MATHEMATICA
| Clear[fQ, fP, lst, n]; fQ[n_]:=PartitionsQ[n]; fP[n_]:=PartitionsP[n]; lst={}; Do[AppendTo[lst, fP[n]-fQ[n]], {n, 5!}]; lst [From Vladimir Orlovsky, Sep 19 2009]
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PROG
| (Pari) x='x+O('x^66); /* that many terms */
Vec(1/eta(x)-eta(x^2)/eta(x)) /* show terms (omitting initial zeros) */
/* Joerg Arndt, Jun 21 2011 */
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CROSSREFS
| A000041(n)-A000009(n).
Cf. A038348.
Sequence in context: A047625 A147871 A004397 * A147955 A134591 A058611
Adjacent sequences: A047964 A047965 A047966 * A047968 A047969 A047970
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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