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Search: id:A116931
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| A116931 |
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Partitions of n in which each part, with the possible exception of the largest, occurs at least twice. |
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2
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| 1, 2, 2, 4, 4, 8, 8, 13, 15, 22, 24, 37, 40, 57, 64, 89, 98, 135, 149, 199, 224, 292, 325, 424, 472, 601, 676, 850, 950, 1191, 1329, 1643, 1845, 2258, 2524, 3082, 3442, 4158, 4659, 5591, 6246, 7472, 8338, 9903, 11072, 13077, 14586, 17184, 19150, 22431
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OFFSET
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1,2
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COMMENT
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Also, partitions of n in which any two distinct parts differ by at least 2. Example: a(5)=4 because we have [5],[4,1],[3,1,1], and [1,1,1,1,1].
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REFERENCES
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P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 52, Article 298.
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FORMULA
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G.f.=sum(x^k*product(1+x^(2j)/(1-x^j), j=1..k-1)/(1-x^k), k=1..infinity). More generally, the g.f. of partitions of n in which each part, with the possible exception of the largest, occurs at least b times, is sum(x^k*product(1+x^(bj)/(1-x^j), j=1..k-1)/(1-x^k), k=1..infinity). It is also the g.f. of partitions of n in which any two distinct parts differ by at least b.
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EXAMPLE
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a(5)=4 because we have [5],[3,1,1],[2,1,1,1], and [1,1,1,1,1].
q + 2*q^2 + 2*q^3 + 4*q^4 + 4*q^5 + 8*q^6 + 8*q^7 + 13*q^8 + 15*q^9 + ...
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MAPLE
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g:=sum(x^k*product(1+x^(2*j)/(1-x^j), j=1..k-1)/(1-x^k), k=1..70): gser:=series(g, x=0, 60): seq(coeff(gser, x^n), n=1..54);
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PROGRAM
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(PARI) {a(n) = if( n<1, 0, polcoeff( sum(k=1, n, x^k / (1 - x^k) * prod(j=1, k-1, 1 + x^(2*j) / (1 - x^j), 1 + x * O(x^(n-k)))), n))} /* Michael Somos Jan 26 2008 */
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CROSSREFS
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Cf. A116932.
Sequence in context: A046971 A051754 A108747 this_sequence A145810 A034397 A082267
Adjacent sequences: A116928 A116929 A116930 this_sequence A116932 A116933 A116934
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 27 2006
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