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

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

0,6

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

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.

Example

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

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 *)

Prog

(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)", "));

Crossrefs

Cf. A000009, A005895, A003406.

Sequence in context: A266769 A008763 A349795 * A147953 A163468 A069183

Adjacent sequences: A005893 A005894 A005895 * A005897 A005898 A005899

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Simon Plouffe

Extensions

More terms from Michael Somos.

Status

approved