A005895 Weighted count of partitions with distinct parts.
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
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
Programs
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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 *) -
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 */
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]
Extensions
More terms from James Sellers, Dec 24 1999
Comments