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 A067659 Number of partitions of n into distinct parts such that number of parts is odd. 61
 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). Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA For g.f. see under A067661. a(n) = (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 A000009(n) = a(n) + A067661(n). - Gus Wiseman, Jan 09 2021 EXAMPLE From Gus Wiseman, Jan 09 2021: (Start) The a(5) = 1 through a(15) = 14 partitions (A-F = 10..15):   5   6     7     8     9     A     B     C     D     E     F       321   421   431   432   532   542   543   643   653   654                   521   531   541   632   642   652   743   753                         621   631   641   651   742   752   762                               721   731   732   751   761   843                                     821   741   832   842   852                                           831   841   851   861                                           921   931   932   942                                                 A21   941   951                                                       A31   A32                                                       B21   A41                                                             B31                                                             C21                                                             54321 (End) 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-François 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 *) Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&OddQ[Length[#]]&]], {n, 0, 30}] (* Gus Wiseman, Jan 09 2021 *) 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 Dominates A000009. Numbers with these strict partitions as binary indices are A000069. The non-strict version is A027193. The Heinz numbers of these partitions are A030059. The even version is A067661. The version for rank is A117193, with non-strict version A101707. The ordered version is A332304, with non-strict version A166444. Other cases of odd length: - A024429 counts set partitions of odd length. - A089677 counts ordered set partitions of odd length. - A174726 counts ordered factorizations of odd length. - A339890 counts factorizations of odd length. A008289 counts strict partitions by sum and length. A026804 counts partitions whose least part is odd, with strict case A026832. Cf. A000700, A027187, A030229, A117192, A332305. Sequence in context: A027197 A332577 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 May 21 14:54 EDT 2022. Contains 353921 sequences. (Running on oeis4.)