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A027349
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Number of partitions of n into distinct odd parts, the least being 1.
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5
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1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 4, 4, 5, 6, 6, 6, 8, 8, 9, 9, 11, 12, 13, 13, 16, 17, 18, 19, 22, 24, 25, 27, 30, 33, 35, 37, 41, 46, 47, 51, 56, 61, 64, 69, 75, 82, 86, 92, 100, 109, 114, 122, 133, 143, 151, 161, 174
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,13
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COMMENTS
| Column 1 of A116860. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 27 2006
Also number of partitions of n such that the largest part occurs exactly once and each number smaller than the largest part occurs an even nonzero number of times. Example: a(17)=3 because we have [3,2,2,2,2,2,2,1,1],[3,2,2,2,2,1,1,1,1,1,1] and [3,2,2,1,1,1,1,1,1,1,1,1,1]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 13 2006
a(n) is the number of symmetric stack polyominoes of area n with square core. The core of stack is the set of all maximal columns. The core is a square when the number of columns is equal to their height. Equivalently, a(n) is the number of symmetric unimodal compositions of n, where the number of the parts of maximum value equal the maximum value itself. For instance, for n = 20, we have the following stacks: (2,4,4,4,4,2), (1,1,4,4,4,4,1,1), (1,1,1,1,1,1,1,1,2,2,1,1,1,1,1,1,1,1). [Emanuele Munarini, Apr 8 2011]
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FORMULA
| G.f.=x*product(1+x^(2i-1), i=2..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 27 2006
G.f.=sum(x^(k^2)/product(1-x^(2j), j=1..k-1), k=1..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 13 2006
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EXAMPLE
| a(17)=3 because we have [13,3,1],[11,5,1] and [9,7,1].
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MAPLE
| N := 100; t1 := series(mul(1+x^(2*k+1), k=1..N), x, N); A027349 := proc(n) coeff(t1, x, n); end;
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MATHEMATICA
| a[n_]:=CoefficientList[Series[1+Sum[x^((k+1)^2)/Product[(1-x^(2i)), {i, 1, k}], {k, 0, n}], {x, 0, n}], x] [Emanuele Munarini, Apr 8 2011]
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CROSSREFS
| Cf. A116860, A001523, A096441, A188674.
Sequence in context: A178696 A161090 A178697 * A053277 A078661 A029263
Adjacent sequences: A027346 A027347 A027348 * A027350 A027351 A027352
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KEYWORD
| nonn
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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