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A005896
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Weighted count of partitions with odd parts.
(Formerly M2338)
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3
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0, 0, 0, 1, 1, 3, 4, 7, 9, 14, 19, 26, 34, 45, 59, 76, 96, 121, 153, 189, 234, 288, 353, 428, 519, 625, 752, 900, 1073, 1274, 1512, 1784, 2101, 2470, 2894, 3382, 3946, 4590, 5330, 6179, 7144, 8246, 9505, 10931, 12552, 14396, 16476, 18831, 21495
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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REFERENCES
| Andrews, George E. Ramanujan's "lost" notebook. V. Euler's partition identity. Adv. in Math. 61 (1986), no. 2, 156-164.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..300
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FORMULA
| G.f.: Sum_{n=0..inf} {S(q)-1/((1-q)(1-q^3)...(1-q^(2n+1)))}, where S(q) = g.f. for A000009.
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MATHEMATICA
| max = 48; f[n_, x_] := Product[ 1/(1-x^(2k+1)), {k, 0, n}]; g[x_] = Sum[ f[max/2, x] - f[n, x], {n, 0, max/2}]; CoefficientList[ Series[ g[x], {x, 0, max}], x] (* From Jean-François Alcover, Nov 17 2011, after g.f. *)
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PROG
| (PARI) /* set "infinity" */ oo = 50; /* G.f. for partitions with odd parts: */ Q(n, q) = prod(k=0, n, 1/(1-q^(2*k+1)), 1+q*O(q^oo)); /* G.f. for A000009: */ Sq = Q(oo/2, q); /* G.f. for A005896: */ Sq0 = sum(n=0, oo/2, Sq-Q(n, q)); for(n=0, 48, print1(polcoeff(Sq0, n)", "))
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CROSSREFS
| Cf. A000009, A005895, A003406.
Sequence in context: A140208 A098390 A008763 * A147953 A163468 A069183
Adjacent sequences: A005893 A005894 A005895 * A005897 A005898 A005899
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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EXTENSIONS
| More terms from Michael Somos.
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