A273225 Number of bipartitions of n wherein odd parts are distinct (and even parts are unrestricted).

1, 2, 3, 6, 11, 18, 28, 44, 69, 104, 152, 222, 323, 460, 645, 902, 1254, 1722, 2343, 3174, 4278, 5722, 7601, 10056, 13250, 17358, 22623, 29382, 38021, 48984, 62857, 80404, 102528, 130282, 165002, 208398, 262495, 329666, 412878, 515840

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Number of bipartitions of 'n' wherein odd parts are distinct (and even parts are unrestricted).

G.f. is the square of the g.f. of A006950. - Vaclav Kotesovec, Mar 25 2017

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

M. D. Hirschhorn and J. A. Sellers, Arithmetic properties of partitions with odd parts distinct, Ramanujan J. 22 (2010), 273--284.

M. S. Mahadeva Naika and D. S. Gireesh, Arithmetic Properties of Partition k-tuples with Odd Parts Distinct, JIS, Vol. 19 (2016), Article 16.5.7

Michael Somos, Introduction to Ramanujan theta functions

L. Wang, Arithmetic properties of partition triples with odd parts distinct, Int. J. Number Theory, 11 (2015), 1791--1805.

L. Wang, Arithmetic properties of partition quadruples with odd parts distinct, Bull. Aust. Math. Soc., doi:10.1017/S0004972715000647.

L. Wang, New congruences for partitions where the odd parts are distinct, J. Integer Seq. (2015), article 15.4.2.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

G.f.: Product_{k>=1} (1 + x^k)^2 / (1 - x^(4*k))^2, corrected by Vaclav Kotesovec, Mar 25 2017

Expansion of 1 / psi(-x)^2 in powers of x where psi() is a Ramanujan theta function.

a(n) ~ exp(Pi*sqrt(n))/(2^(5/2)*n^(5/4)). - Vaclav Kotesovec, Jul 05 2016

Euler transform of period 4 sequence [2, 0, 2, 2, ...]. - Michael Somos, Mar 02 2019

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = 2 * exp(-Pi / 4) * sqrt(2) * Gamma(3/4)^2 / sqrt(Pi) = A388950. - Simon Plouffe, Sep 21 2025

Example

a(4)=11 because "(0,4)=(0,3+1)=(0,2+2)=(1,3)=(1,2+1)=(2,2)=(4,0)=(3+1,0)=(2+2,0)=(3,1)=(2+1,1)".
G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 11*x^4 + 18*x^5 + 28*x^6 + 44*x^7 + ... - Michael Somos, Mar 02 2019
G.f. = q^-1 + 2*q^3 + 3*q^7 + 6*q^11 + 11*q^15 + 18*q^19 + 28*q^23 + ... - Michael Somos, Mar 02 2019

Maple

Digits:=200:with(PolynomialTools): with(qseries): with(ListTools):
GenFun:=series(etaq(q, 2, 100)^2/etaq(q, 1, 100)^2/etaq(q, 4, 100)^2, q, 50):
CoefficientList(sort(convert(GenFun, polynom), q, ascending), q);

Mathematica

s = QPochhammer[-1, x]^2/(4*QPochhammer[x^4, x^4]^2) + O[x]^40; CoefficientList[s, x] (* Jean-François Alcover, May 20 2016 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2, x^4] / QPochhammer[ x])^2, {x, 0, n}]; (* Michael Somos, Mar 02 2019 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0 , A = x * O(x^n); polcoeff( eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A))^2, n))}; /* Michael Somos, Mar 02 2019 */

Crossrefs

For a version with signs see A274621.

Cf. A006950.

Sequence in context: A131512 A147388 A180712 * A274621 A291725 A003479

Adjacent sequences: A273222 A273223 A273224 * A273226 A273227 A273228

Keyword

nonn

Author

M.S. Mahadeva Naika, May 18 2016

Status

approved