OFFSET
0,3
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-conjuage 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
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.
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
EXAMPLE
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 + ...
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 *)
PROG
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Naohiro Nomoto, Feb 24 2002
EXTENSIONS
More terms from Emeric Deutsch, Feb 10 2006
STATUS
approved