cp's OEIS Frontend

A067357 Number of self-conjugate partitions of 4n+1 into odd parts.

%I A067357 #70 Aug 11 2025 10:21:37
%S A067357 1,1,2,2,2,3,4,4,5,6,6,8,10,10,12,14,15,18,20,22,26,29,32,36,40,44,50,
%T A067357 56,60,68,76,82,92,101,110,122,134,146,160,176,191,210,230,248,272,
%U A067357 296,320,350,380,410,446,484,522,566,612,660,715,772,830,896,966,1038,1120
%N A067357 Number of self-conjugate partitions of 4n+1 into odd parts.
%C A067357 Also number of partitions of n in which even parts are distinct and if k occurs then so does every positive even number less than k (Dean Hickerson). Absolute values of the terms of A053254. - _Emeric Deutsch_, Feb 10 2006
%C A067357 The number of self-conjugate partitions of n into odd parts is nonzero if and only if n = 4*k + 1 for some nonnegative integer k. - _Michael Somos_, Jul 25 2015
%C A067357 Also number of C3v plane partitions of n = 3*k + 1 with rank 1 ; equivalently number of self-conjugate integer partitions with (weight-length) = n. - _Wouter Meeussen_, May 23 2025
%D A067357 P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p. 260, Article 512.
%H A067357 G. C. Greubel, <a href="/A067357/b067357.txt">Table of n, a(n) for n = 0..1000</a>
%H A067357 A. S. Andersen, <a href="https://arxiv.org/abs/2007.09794">A Bijection Between Two Different Classes of Partitions Enumerated by p_nu(n)</a>, arXiv:2007.09794 [math.CO], 2020.
%H A067357 George E. Andrews, <a href="https://georgeandrews1.github.io/pdf/320.pdf">The Bhargava-Adiga Summation and Partitions</a>, Journal of the Indian Mathematical Society, Volume 84, Issue 3-4, 2017.
%H A067357 George E. Andrews, <a href="https://georgeandrews1.github.io/pdf/323.pdf">Integer partitions with even parts below odd parts and the mock theta functions</a>, to appear in Annals of Combinatorics, 2017.
%F A067357 G.f.: Sum_{k>=0} q^(k*(k+1)) / ((1-q) * (1-q^3) ... (1-q^(2*k+1))). - _Emeric Deutsch_ and _Dean Hickerson_
%F A067357 G.f.: Sum_{k>=0} q^k * (1+q) * (1+q^3) ... (1+q^(2*k-1)). - _Dean Hickerson_ and _Vladeta Jovovic_
%F A067357 G.f.: 1/(1 - x*(1+x)/(1 + x^2*(1-x^2)/(1 - x^3*(1+x^3)/(1 + x^4*(1-x^4)/(1 - x^5*(1+x^5)/(1 - ...)))))), a continued fraction. - _Paul D. Hanna_, Jul 09 2013
%F A067357 From _Michael Somos_, Jul 25 2015: (Start)
%F A067357 Expansion of nu(-x) in powers of x where nu() is a 3rd-order mock theta function.
%F A067357 a(n) = (-1)^n * A053254(n).
%F A067357 a(2*n) = A085140(n).
%F A067357 a(2*n + 1) = A053253(n). (End)
%F A067357 a(n) ~ exp(Pi*sqrt(n/6)) / (2^(3/2)*sqrt(n)). - _Vaclav Kotesovec_, Jun 15 2019
%e A067357 a(5)=3 because we have [11,1,1,1,1,1,1,1,1,1,1], [9,3,3,1,1,1,1,1,1] and [5,5,5,3,3].
%e A067357 G.f. = 1 + x + 2*x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 4*x^7 + 5*x^8 + ...
%p A067357 g:=sum(q^(k*(k+1))/product(1-q^(2*j+1),j=0..k),k=0..8): gser:=series(g,q=0,80): seq(coeff(gser,q,n),n=0..75); # _Emeric Deutsch_, Feb 10 2006
%t A067357 a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(k^2 + k) / Product[ 1 - x^i, {i, 1, 2 k + 1, 2}], {k, 0, (Sqrt[ 4 n + 1] - 1) / 2}], {x, 0, n}]]; (* _Michael Somos_, Jul 25 2015 *)
%t A067357 Table[Length[Flatten[Table[Select[IntegerPartitions[w], (w-Length[#])== r && TransposePartition[#] == # &],{w,r,1+2r}],1]],{r,1,17}] (* _Wouter Meeussen_, May 24 2025 *)
%o A067357 (PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint( 4*n+1) -1) \ 2, x^(k^2 + k) / prod(j=0, k, 1 - x^(2*j+1), 1 + x * O(x^(n - k^2 - k)))), n))}; /* _Michael Somos_, Jan 27 2008 */
%o A067357 (PARI) /* Continued Fraction Expansion: */
%o A067357 {a(n)=local(CF); CF=1+x; for(k=0, n, CF=1/(1 + (-x)^(n-k+1)*(1 - (-x)^(n-k+1))*CF+x*O(x^n))); polcoeff(CF, n)} \\ _Paul D. Hanna_, Jul 09 2013
%Y A067357 Cf. A000700, A000701.
%Y A067357 Cf. A053253, A053254, A085140, A047993.
%K A067357 easy,nonn
%O A067357 0,3
%A A067357 _Naohiro Nomoto_, Feb 24 2002
%E A067357 More terms from _Emeric Deutsch_, Feb 10 2006