A101277 Number of partitions of 2n in which all odd parts occur with multiplicity 2. There is no restriction on the even parts.

1, 2, 3, 6, 10, 16, 25, 38, 57, 84, 121, 172, 243, 338, 465, 636, 862, 1158, 1546, 2050, 2702, 3542, 4616, 5986, 7729, 9932, 12707, 16196, 20563, 26010, 32788, 41194, 51591, 64418, 80195, 99558, 123269, 152226, 187514, 230434, 282519, 345596, 421844, 513834

Offset

0,2

Comments

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

This is also A080054 times 1/Product_{k>=1} (1 - x^(2k)).

There are no partitions of 2n+1 in which all odd parts occur with multiplicity 2. - Michael Somos, Oct 27 2008

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cristina Ballantine, Mircea Merca, Jacobi's Four and Eight Squares Theorems and Partitions into Distinct Parts, Mediterranean Journal of Mathematics (2019) Vol. 16, No. 2, 26.

Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.

Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, q-Pochhammer Symbol, Ramanujan Theta Functions

Formula

Euler transform of period 4 sequence [2, 0, 2, 1, ...]. - Michael Somos, Feb 10 2005

G.f.: (1/theta_4(0, x))*Product_{k>0}(1+x^(2k)) = theta_4(0, x^2)/theta_4(0, x)*Product_{k>0}(1-x^(2k)) = 1/Product_{k>0} ((1-x^(2k-1))^2 * (1-x^(4k))).

Expansion of 1 / (psi(-x) * chi(-x)) in powers of x where psi(), chi() are Ramanujan theta functions. - Michael Somos, Oct 27 2008

Expansion of q^(1/12) * eta(q^2)^2 / (eta(q)^2 * eta(q^4)) in powers of q. - Michael Somos, Oct 27 2008

a(n) ~ sqrt(5) * exp(Pi*sqrt(5*n/6)) / (8*sqrt(3)*n). - Vaclav Kotesovec, Aug 30 2015

G.f.: 2/((x; x)_inf * (-1; -x)_inf), where (a; q)_inf is the q-Pochhammer symbol. - Vladimir Reshetnikov, Nov 22 2016

Expansion of phi(-x^2) / f(-x)^2 = chi(x) / f(-x) = 1 / (chi(-x)^2 * f(-x^4)) = f(-x^4) / psi(-x)^2 = psi(-x) / chi(-x) = chi(x)^2 / psi(-x^2) in powers of x. - Michael Somos, Nov 22 2016

Example

G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 25*x^6 + 38*x^7 + 57*x^8 + ...
G.f. = 1/q + 2*q^11 + 3*q^23 + 6*q^35 + 10*q^47 + 16*q^59 + 25*q^71 + ...
E.g. 12 = 10 + 2 = 10 + 1 + 1 = 8 + 4 = 8 + 2 + 2 = 8 + 2 + 1 + 1 = 6 + 6 = 6 + 4 + 2 = 6 + 4 + 1 + 1 = 6 + 3 + 3 = 6 + 2 + 2 + 2 = 6 + 2 + 2 + 1 + 1 = 5 + 5 + 2 = 5 + 5 + 1 + 1 = 4 + 4 + 4 = 4 + 4 + 2 + 2 = 4 + 4 + 2 + 1 + 1 = 4 + 3 + 3 + 2 = 4 + 3 + 3 + 1 + 1 = 4 + 2 + 2 + 2 + 2 = 4 + 2 + 2 + 2 + 1 + 1 = 3 + 3 + 2 + 2 + 2 = 3 + 3 + 2 + 2 + 1 + 1 = 2 + 2 + 2 + 2 + 2 + 2 = 2 + 2 + 2 + 2 + 2 + 1 + 1.

Maple

series(product(1/((1-x^(2*k-1))^2*(1-x^(4*k))), k=1..100), x=0, 100);

Mathematica

nmax=50; CoefficientList[Series[Product[1/((1-x^(2*k-1))^2 * (1-x^(4*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 30 2015 *)
(2/(QPochhammer[x] QPochhammer[-1, -x]) + O[x]^45)[[3]] (* Vladimir Reshetnikov, Nov 22 2016 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^2] / QPochhammer[ x]^2, {x, 0, n}]; (* Michael Somos, Nov 22 2016 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Nov 22 2016 *)
a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x, -x] QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, Nov 22 2016 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)), n))}; /* Michael Somos, Feb 10 2005 */

Crossrefs

Cf. A015128, A098151, A080054.

Sequence in context: A260599 A280908 A146163 * A262984 A201077 A355383

Adjacent sequences: A101274 A101275 A101276 * A101278 A101279 A101280

Keyword

nonn

Author

Noureddine Chair, Dec 20 2004; revised Jan 05 2005

Status

approved