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

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, 56, 60, 68, 76, 82, 92, 101, 110, 122, 134, 146, 160, 176, 191, 210, 230, 248, 272, 296, 320, 350, 380, 410, 446, 484, 522, 566, 612, 660, 715, 772, 830, 896, 966, 1038, 1120

Offset: 0

Author

Naohiro Nomoto, Feb 24 2002

Keywords

• easy
• nonn

Comments

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

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

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

Examples

			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].
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 + ...
		

References

• 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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• A. S. Andersen, A Bijection Between Two Different Classes of Partitions Enumerated by p_nu(n), arXiv:2007.09794 [math.CO], 2020.
• George E. Andrews, The Bhargava-Adiga Summation and Partitions, Journal of the Indian Mathematical Society, Volume 84, Issue 3-4, 2017.
• George E. Andrews, Integer partitions with even parts below odd parts and the mock theta functions, to appear in Annals of Combinatorics, 2017.

Crossrefs

Cf. A000700, A000701.

Cf. A053253, A053254, A085140, A047993.

Programs

Maple

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

Mathematica

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 *)
Table[Length[Flatten[Table[Select[IntegerPartitions[w], (w-Length[#])== r && TransposePartition[#] == # &],{w,r,1+2r}],1]],{r,1,17}] (* Wouter Meeussen, May 24 2025 *)

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 */

PARI

/* Continued Fraction Expansion: */
{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

Formula

G.f.: Sum_{k>=0} q^(k*(k+1)) / ((1-q) * (1-q^3) ... (1-q^(2*k+1))). - Emeric Deutsch and Dean Hickerson

G.f.: Sum_{k>=0} q^k * (1+q) * (1+q^3) ... (1+q^(2*k-1)). - Dean Hickerson and Vladeta Jovovic

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

From Michael Somos, Jul 25 2015: (Start)

Expansion of nu(-x) in powers of x where nu() is a 3rd-order mock theta function.

a(n) = (-1)^n * A053254(n).

a(2*n) = A085140(n).

a(2*n + 1) = A053253(n). (End)

a(n) ~ exp(Pi*sqrt(n/6)) / (2^(3/2)*sqrt(n)). - Vaclav Kotesovec, Jun 15 2019

Extensions

More terms from Emeric Deutsch, Feb 10 2006