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

%I A273225 #38 Mar 19 2025 13:20:25
%S A273225 1,2,3,6,11,18,28,44,69,104,152,222,323,460,645,902,1254,1722,2343,
%T A273225 3174,4278,5722,7601,10056,13250,17358,22623,29382,38021,48984,62857,
%U A273225 80404,102528,130282,165002,208398,262495,329666,412878,515840
%N A273225 Number of bipartitions of n wherein odd parts are distinct (and even parts are unrestricted).
%C A273225 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A273225 Number of bipartitions of 'n' wherein odd parts are distinct (and even parts are unrestricted).
%C A273225 G.f. is the square of the g.f. of A006950. - _Vaclav Kotesovec_, Mar 25 2017
%H A273225 Vaclav Kotesovec, <a href="/A273225/b273225.txt">Table of n, a(n) for n = 0..2000</a>
%H A273225 M. D. Hirschhorn and J. A. Sellers, <a href="http://dx.doi.org/10.1007/s11139-010-9225-6">Arithmetic properties of partitions with odd parts distinct</a>, Ramanujan J. 22 (2010), 273--284.
%H A273225 M. S. Mahadeva Naika and D. S. Gireesh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Naika/naika2.html">Arithmetic Properties of Partition k-tuples with Odd Parts Distinct</a>, JIS, Vol. 19 (2016), Article 16.5.7
%H A273225 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A273225 L. Wang, <a href="http://dx.doi.org/10.1142/S1793042115500773">Arithmetic properties of partition triples with odd parts distinct</a>, Int. J. Number Theory, 11 (2015), 1791--1805.
%H A273225 L. Wang, <a href="http://dx.doi.org/10.1017/S0004972715000647">Arithmetic properties of partition quadruples with odd parts distinct</a>, Bull. Aust. Math. Soc., doi:10.1017/S0004972715000647.
%H A273225 L. Wang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Wang2/wang31.html">New congruences for partitions where the odd parts are distinct</a>, J. Integer Seq. (2015), article 15.4.2.
%H A273225 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A273225 G.f.: Product_{k>=1} (1 + x^k)^2 / (1 - x^(4*k))^2, corrected by _Vaclav Kotesovec_, Mar 25 2017
%F A273225 Expansion of 1 / psi(-x)^2 in powers of x where psi() is a Ramanujan theta function.
%F A273225 a(n) ~ exp(Pi*sqrt(n))/(2^(5/2)*n^(5/4)). - _Vaclav Kotesovec_, Jul 05 2016
%F A273225 Euler transform of period 4 sequence [2, 0, 2, 2, ...]. - _Michael Somos_, Mar 02 2019
%e A273225 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)".
%e A273225 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
%e A273225 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
%p A273225 Digits:=200:with(PolynomialTools): with(qseries): with(ListTools):
%p A273225 GenFun:=series(etaq(q,2,100)^2/etaq(q,1,100)^2/etaq(q,4,100)^2,q,50):
%p A273225 CoefficientList(sort(convert(GenFun,polynom),q,ascending),q);
%t A273225 s = QPochhammer[-1, x]^2/(4*QPochhammer[x^4, x^4]^2) + O[x]^40; CoefficientList[s, x] (* _Jean-François Alcover_, May 20 2016 *)
%t A273225 a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2, x^4] / QPochhammer[ x])^2, {x, 0, n}]; (* _Michael Somos_, Mar 02 2019 *)
%o A273225 (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 */
%Y A273225 For a version with signs see A274621.
%Y A273225 Cf. A006950.
%K A273225 nonn
%O A273225 0,2
%A A273225 _M.S. Mahadeva Naika_, May 18 2016