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 A005896 Weighted count of partitions with odd parts. (Formerly M2338) 3

%I M2338

%S 0,0,0,1,1,3,4,7,9,14,19,26,34,45,59,76,96,121,153,189,234,288,353,

%T 428,519,625,752,900,1073,1274,1512,1784,2101,2470,2894,3382,3946,

%U 4590,5330,6179,7144,8246,9505,10931,12552,14396,16476,18831,21495

%N Weighted count of partitions with odd parts.

%D Andrews, George E. Ramanujan's "lost" notebook. V. Euler's partition identity. Adv. in Math. 61 (1986), no. 2, 156-164.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A005896/b005896.txt">Table of n, a(n) for n = 0..300</a>

%F 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.

%e G.f. = x^3 + x^4 + 3*x^5 + 4*x^6 + 7*x^7 + 9*x^8 + 14*x^9 + 19*x^10 + ... - _Michael Somos_, Oct 21 2018

%t 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] (* _Jean-François Alcover_, Nov 17 2011, after g.f. *)

%t a[ n_] := With[{A = 1 / QPochhammer[ q, q^2]}, SeriesCoefficient[ Sum[A - 1 / QPochhammer[ q, q^2, k], {k, 1, n/2}], {q, 0, n}]]; (* _Michael Somos_, Oct 21 2018 *)

%o (PARI) /* set maximum */ MM = 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^MM))); /* G.f. for A000009: */ Sq = Q(MM/2, q); /* G.f. for A005896: */ Sq0 = sum(n=0, MM/2, Sq-Q(n, q)); for(n=0, 48, print1(polcoeff(Sq0, n)","));

%Y Cf. A000009, A005895, A003406.

%K nonn,easy,nice

%O 0,6

%A _N. J. A. Sloane_, _Simon Plouffe_

%E More terms from Michael Somos.

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Last modified March 21 04:59 EDT 2019. Contains 321364 sequences. (Running on oeis4.)