A005896 Weighted count of partitions with odd parts.

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

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy
• nice

Examples

			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
		

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).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A000009, A005895, A003406.

Programs

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] (* Jean-François Alcover, Nov 17 2011, after g.f. *)
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 *)

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)","));

Formula

G.f.: Sum_{n=0..infinity} {S(q)-1/((1-q)(1-q^3)...(1-q^(2n+1)))}, where S(q) = g.f. for A000009.

Extensions

More terms from Michael Somos.