This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A027193 #77 Apr 29 2023 13:59:17
%S A027193 0,1,1,2,2,4,5,8,10,16,20,29,37,52,66,90,113,151,190,248,310,400,497,
%T A027193 632,782,985,1212,1512,1851,2291,2793,3431,4163,5084,6142,7456,8972,
%U A027193 10836,12989,15613,18646,22316,26561,31659,37556,44601,52743,62416,73593,86809,102064,120025,140736
%N A027193 Number of partitions of n into an odd number of parts.
%C A027193 Number of partitions of n in which greatest part is odd.
%C A027193 Number of partitions of n+1 into an even number of parts, the least being 1. Example: a(5)=4 because we have [5,1], [3,1,1,1], [2,1,1] and [1,1,1,1,1,1].
%C A027193 Also number of partitions of n+1 such that the largest part is even and occurs only once. Example: a(5)=4 because we have [6], [4,2], [4,1,1] and [2,1,1,1,1]. - _Emeric Deutsch_, Apr 05 2006
%C A027193 Also the number of partitions of n such that the number of odd parts and the number of even parts have opposite parities. Example: a(8)=10 is a count of these partitions: 8, 611, 521, 431, 422, 41111, 332, 32111, 22211, 2111111. - _Clark Kimberling_, Feb 01 2014, corrected Jan 06 2021
%C A027193 In Chaves 2011 see page 38 equation (3.20). - _Michael Somos_, Dec 28 2014
%C A027193 Suppose that c(0) = 1, that c(1), c(2), ... are indeterminates, that d(0) = 1, and that d(n) = -c(n) - c(n-1)*d(1) - ... - c(0)*d(n-1). When d(n) is expanded as a polynomial in c(1), c(2),..,c(n), the terms are of the form H*c(i_1)*c(i_2)*...*c(i_k). Let P(n) = [c(i_1), c(i_2), ..., c(i_k)], a partition of n. Then H is negative if P has an odd number of parts, and H is positive if P has an even number of parts. That is, d(n) has A027193(n) negative coefficients, A027187(n) positive coefficients, and A000041 terms. The maximal coefficient in d(n), in absolute value, is A102462(n). - _Clark Kimberling_, Dec 15 2016
%D A027193 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 39, Example 7.
%H A027193 T. D. Noe, <a href="/A027193/b027193.txt">Table of n, a(n) for n = 0..999</a>
%H A027193 Roland Bacher and P. De La Harpe, <a href="https://hal.archives-ouvertes.fr/hal-01285685/document">Conjugacy growth series of some infinitely generated groups</a>, hal-01285685v2, 2016.
%H A027193 D. R. C. Chaves, <a href="http://hdl.handle.net/10183/29250">Um estudo combinatório e comparativo de identidades teta parciais de Andrews e Ramanujan</a>, 2011. In Portuguese.
%H A027193 Mircea Merca, <a href="http://dx.doi.org/10.1016/j.jnt.2015.08.014">Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer</a>, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function p_0(n).
%F A027193 a(n) = (A000041(n) - (-1)^n*A000700(n)) / 2.
%F A027193 For g.f. see under A027187.
%F A027193 G.f.: Sum(k>=1, x^(2*k-1)/Product(j=1..2*k-1, 1-x^j ) ). - _Emeric Deutsch_, Apr 05 2006
%F A027193 G.f.: - Sum(k>=1, (-x)^(k^2)) / Product(k>=1, 1-x^k ). - _Joerg Arndt_, Feb 02 2014
%F A027193 G.f.: Sum(k>=1, x^(k*(2*k-1)) / Product(j=1..2*k, 1-x^j)). - _Michael Somos_, Dec 28 2014
%F A027193 a(2*n) = A000701(2*n), a(2*n-1) = A046682(2*n-1); a(n) = A000041(n)-A027187(n). - _Reinhard Zumkeller_, Apr 22 2006
%e A027193 G.f. = x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 5*x^6 + 8*x^7 + 10*x^8 + 16*x^9 + 20*x^10 + ...
%e A027193 From _Gus Wiseman_, Feb 11 2021: (Start)
%e A027193 The a(1) = 1 through a(8) = 10 partitions into an odd number of parts are the following. The Heinz numbers of these partitions are given by A026424.
%e A027193 (1) (2) (3) (4) (5) (6) (7) (8)
%e A027193 (111) (211) (221) (222) (322) (332)
%e A027193 (311) (321) (331) (422)
%e A027193 (11111) (411) (421) (431)
%e A027193 (21111) (511) (521)
%e A027193 (22111) (611)
%e A027193 (31111) (22211)
%e A027193 (1111111) (32111)
%e A027193 (41111)
%e A027193 (2111111)
%e A027193 The a(1) = 1 through a(8) = 10 partitions whose greatest part is odd are the following. The Heinz numbers of these partitions are given by A244991.
%e A027193 (1) (11) (3) (31) (5) (33) (7) (53)
%e A027193 (111) (1111) (32) (51) (52) (71)
%e A027193 (311) (321) (322) (332)
%e A027193 (11111) (3111) (331) (521)
%e A027193 (111111) (511) (3221)
%e A027193 (3211) (3311)
%e A027193 (31111) (5111)
%e A027193 (1111111) (32111)
%e A027193 (311111)
%e A027193 (11111111)
%e A027193 (End)
%p A027193 g:=sum(x^(2*k)/product(1-x^j,j=1..2*k-1),k=1..40): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=1..45); # _Emeric Deutsch_, Apr 05 2006
%t A027193 nn=40;CoefficientList[Series[ Sum[x^(2j+1)Product[1/(1- x^i),{i,1,2j+1}],{j,0,nn}],{x,0,nn}],x] (* _Geoffrey Critzer_, Dec 01 2012 *)
%t A027193 a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[ n], OddQ[ Length@#] &]]; (* _Michael Somos_, Dec 28 2014 *)
%t A027193 a[ n_] := If[ n < 1, 0, Length@Select[ IntegerPartitions[ n], OddQ[ First@#] &]]; (* _Michael Somos_, Dec 28 2014 *)
%t A027193 a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[ n + 1], #[[-1]] == 1 && EvenQ[ Length@#] &]]; (* _Michael Somos_, Dec 28 2014 *)
%t A027193 a[ n_] := If[ n < 1, 0, Length@Select[ IntegerPartitions[ n + 1], EvenQ[ First@#] && (Length[#] < 2 || #[[1]] != #[[2]]) &]]; (* _Michael Somos_, Dec 28 2014 *)
%o A027193 (PARI) {a(n) = if( n<1, 0, polcoeff( sum( k=1, n, if( k%2, x^k / prod( j=1, k, 1 - x^j, 1 + x * O(x^(n-k)) ))), n))}; /* _Michael Somos_, Jul 24 2012 */
%o A027193 (PARI) q='q+O('q^66); concat([0], Vec( (1/eta(q)-eta(q)/eta(q^2))/2 ) ) \\ _Joerg Arndt_, Mar 23 2014
%Y A027193 Cf. A000041, A000700, A000701, A026837, A046682.
%Y A027193 The Heinz numbers of these partitions are A026424 or A244991.
%Y A027193 The even-length version is A027187.
%Y A027193 The case of odd sum as well as length is A160786, ranked by A340931.
%Y A027193 The case of odd maximum as well as length is A340385.
%Y A027193 Other cases of odd length:
%Y A027193 - A024429 counts set partitions of odd length.
%Y A027193 - A067659 counts strict partitions of odd length.
%Y A027193 - A089677 counts ordered set partitions of odd length.
%Y A027193 - A166444 counts compositions of odd length.
%Y A027193 - A174726 counts ordered factorizations of odd length.
%Y A027193 - A332304 counts strict compositions of odd length.
%Y A027193 - A339890 counts factorizations of odd length.
%Y A027193 A000009 counts partitions into odd parts, ranked by A066208.
%Y A027193 A026804 counts partitions whose least part is odd.
%Y A027193 A058695 counts partitions of odd numbers, ranked by A300063.
%Y A027193 A072233 counts partitions by sum and length.
%Y A027193 A101707 counts partitions of odd positive rank.
%Y A027193 Cf. A078408, A174725, A236914, A300272, A340831.
%K A027193 nonn
%O A027193 0,4
%A A027193 _Clark Kimberling_