A005895 Weighted count of partitions with distinct parts. (Formerly M1337)

1, 2, 5, 7, 12, 18, 26, 35, 50, 67, 88, 116, 149, 191, 245, 306, 381, 477, 585, 718, 880, 1067, 1288, 1555, 1863, 2226, 2656, 3151, 3726, 4406, 5180, 6077, 7124, 8316, 9691, 11278, 13080, 15146, 17517, 20204, 23264, 26759, 30705, 35182, 40274, 46000, 52473, 59795, 68018, 77279, 87711, 99395, 112508

Offset

1,2

Comments

Also sum of largest parts of all partitions of n into distinct parts. - Vladeta Jovovic, Feb 15 2004

References

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

S.-Y. Kang, Generalizations of Ramanujan's reciprocity theorem..., J. London Math. Soc., 75 (2007), 18-34. See Eq. (1.5) but beware errors.

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

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Formula

G.f.: sum(n>=0, S(q) - prod(k=1..n, 1+q^k) ), where S(q)=prod(k>=1, 1+q^k) (g.f. for A000009).

G.f. sum(k>=0, (k+1)*x^(k+1) * prod(j=1..k, 1+x^j) ). [Joerg Arndt, Sep 17 2012]

Maple

M:=201; add( mul( (1+q^j), j=1..M) - mul( (1+q^j), j=1..n), n=0..M);
# second Maple program:
b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0, `if`(
      n=0, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, min(n-i, i-1)))))
    end:
a:= n-> add(j*b(n-j, min(n-j, j-1)), j=1..n):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 03 2016

Mathematica

m = 46; f[q_] :=  Sum[ Product[ (1+q^j), {j, 1, m}] - Product[ (1+q^j), {j, 1, n}], {n, 0, m}]; CoefficientList[ f[q], q][[2 ;; m+1]] (* Jean-François Alcover, Apr 13 2012, after Maple *)

Prog

(PARI)
N=66;  x='x+O('x^N);
S=prod(k=1, N, 1+x^k); gf=sum(n=0, N, S-prod(k=1, n, 1+x^k));
/* alternative: Arndt's g.f.: */
/* gf=sum(k=0, N, (k+1)*x^(k+1) * prod(j=1, k, 1+x^j) ); */
Vec(gf)
/* Joerg Arndt, Sep 17 2012 */

Crossrefs

Cf. A005896, A003406, A336902, A336903.

Sequence in context: A023668 A023564 A173088 * A238661 A135525 A319142

Adjacent sequences: A005892 A005893 A005894 * A005896 A005897 A005898

Keyword

nonn,easy,nice

Author

N. J. A. Sloane and Simon Plouffe

Extensions

More terms from James A. Sellers, Dec 24 1999

Status

approved