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A026832
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Number of partitions of n into distinct parts, the least being odd.
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4
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1, 0, 2, 1, 2, 2, 4, 4, 5, 6, 8, 10, 12, 14, 18, 21, 24, 30, 36, 42, 50, 58, 68, 80, 93, 108, 126, 146, 168, 194, 224, 256, 294, 336, 384, 439, 500, 568, 646, 732, 828, 938, 1060, 1194, 1348, 1516, 1704, 1916, 2149, 2408, 2698, 3018, 3372
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OFFSET
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1,3
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COMMENTS
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Fine's numbers L(n).
Also number of partitions of n such that if k is the largest part, then k occurs an odd number of times and each of the numbers 1,2,...,k-1 occurs at least once. Example: a(7)=4 because we have [3,2,1,1],[2,2,2,1],[2,1,1,1,1,1] and [1,1,1,1,1,1,1] - Emeric Deutsch, Mar 29 2006
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REFERENCES
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N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 56, Eq. (26.28).
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LINKS
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Table of n, a(n) for n=1..53.
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FORMULA
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G.f.: Sum_{k>=1} ((-1)^(k+1)*(-1+Product_{i>=k} (1+x^i))). - Vladeta Jovovic, Aug 26 2003
G.f.: Sum_{ k >= 1 } x^(k*(k+1)/2)/((1+x^k)*Product_{i=1..k} (1-x^i) ). - Vladeta Jovovic, Aug 10 2004
(1 + Sum_{n >= 1} a(n)q^n )*(1 + 2 Sum_{m>=1} (-1)^m*q^(m^2)) = Sum_{n >= 1} (-1)^n*q^((3*n^2+n)/2)/(1+q^n). [Fine]
G.f.=sum(x^(2k-1)*product(1+x^j, j=2k..infinity), k=1..infinity). - Emeric Deutsch, Mar 29 2006
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EXAMPLE
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a(7)=4 because we have [7],[6,1],[4,3] and [4,2,1].
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MAPLE
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g:=sum(x^(2*k-1)*product(1+x^j, j=2*k..60), k=1..60): gser:=series(g, x=0, 55); seq(coeff(gser, x^n), n=1..53); - Emeric Deutsch, Mar 29 2006
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MATHEMATICA
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mx=53; Rest[CoefficientList[Series[Sum[x^(2*k-1) Product[1+x^j, {j, 2*k, mx}], {k, mx}], {x, 0, mx}], x]] (* From Jean-François Alcover, Apr 05 2011, after Emeric Deutsch *)
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PROG
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(Haskell)
a026832 n = p 1 n where
p _ 0 = 1
p k m = if m < k then 0 else p (k+1) (m-k) + p (k+1+0^(n-m)) m
-- Reinhard Zumkeller, Jun 14 2012
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CROSSREFS
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Cf. A026804, A026805, A026807, A092265, A096661, A097042.
Cf. A000009, A096765.
Sequence in context: A204596 A163373 A117193 * A225044 A193691 A089408
Adjacent sequences: A026829 A026830 A026831 * A026833 A026834 A026835
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KEYWORD
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nonn,nice,changed
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AUTHOR
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Clark Kimberling
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EXTENSIONS
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More terms from Emeric Deutsch, Mar 29 2006
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STATUS
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approved
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