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A067659 Number of partitions of n into distinct parts such that number of parts is odd. 27
0, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 16, 19, 23, 27, 32, 38, 44, 52, 61, 71, 82, 96, 111, 128, 148, 170, 195, 224, 256, 293, 334, 380, 432, 491, 557, 630, 713, 805, 908, 1024, 1152, 1295, 1455, 1632, 1829, 2048, 2291, 2560, 2859, 3189, 3554, 3958, 4404 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

Ramanujan theta functions: phi(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Joerg Arndt, Matters Computational (The Fxtbook), end of section 16.4.2 "Partitions into distinct parts", pp.348ff

Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function q_o(n).

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

For g.f. see under A067661.

(A000009(n)-A010815(n))/2. - Vladeta Jovovic, Feb 24 2002

Expansion of (1-phi(-q))/(2*chi(-q)) in powers of q where phi(),chi() are Ramanujan theta functions. - Michael Somos, Feb 14 2006

G.f.: sum(n>=1, q^(2*n^2-n) / prod(k=1..2*n-1, 1-q^k ) ). [Joerg Arndt, Apr 01 2014]

a(n) = A067661(n) + A010815(n). - Andrey Zabolotskiy, Apr 12 2017

MAPLE

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,

      `if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))

    end:

a:= n-> b(n$2, 0):

seq(a(n), n=0..80);  # Alois P. Heinz, Apr 01 2014

MATHEMATICA

b[n_, i_, t_] := b[n, i, t] = If[n > i*(i + 1)/2, 0, If[n == 0, t, Sum[b[n - i*j, i - 1, Abs[t - j]], {j, 0, Min[n/i, 1]}]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-Fran├žois Alcover, Jan 16 2015, after Alois P. Heinz *)

CoefficientList[Normal[Series[(QPochhammer[-x, x]-QPochhammer[x])/2, {x, 0, 100}]], x] (* Andrey Zabolotskiy, Apr 12 2017 *)

PROG

(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)/eta(x+A) - eta(x+A))/2, n))} /* Michael Somos, Feb 14 2006 */

(PARI) N=66;  q='q+O('q^N);  S=1+2*sqrtint(N);

gf=sum(n=1, S, (n%2!=0) * q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) );

concat( [0], Vec(gf) )  /* Joerg Arndt, Oct 20 2012 */

(PARI) N=66;  q='q+O('q^N);  S=1+sqrtint(N);

gf=sum(n=1, S, q^(2*n^2-n) / prod(k=1, 2*n-1, 1-q^k ) );

concat( [0], Vec(gf) )  \\ Joerg Arndt, Apr 01 2014

CROSSREFS

Cf. A067661 (even number of parts).

Sequence in context: A237757 A027197 A137793 * A261772 A153156 A017852

Adjacent sequences:  A067656 A067657 A067658 * A067660 A067661 A067662

KEYWORD

easy,nonn

AUTHOR

Naohiro Nomoto, Feb 23 2002

STATUS

approved

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Last modified August 20 20:23 EDT 2018. Contains 313927 sequences. (Running on oeis4.)