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A100684
Number of partitions of 2n free of multiples of 8 such that 4 occurs at most once. All odd parts occur with even multiplicities. There is no restriction on the other even parts.
1
1, 2, 4, 8, 12, 20, 32, 48, 72, 106, 152, 216, 305, 422, 580, 792, 1068, 1432, 1908, 2520, 3313, 4332, 5628, 7280, 9373, 12008, 15324, 19480, 24661, 31112, 39120, 49016, 61229, 76260, 94692, 117264, 144834, 178412, 219244, 268784, 328746
OFFSET
0,2
LINKS
Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.
FORMULA
G.f.: (1-x^4)*Product((1+x^(2*i))/(1-x^(2*i-1))^2, i=1..infinity). [Vladeta Jovovic]
Expansion of (1 - q^4) * q^(-1/6) * eta(q^4) * eta(q^2) / eta(q)^2 in powers of q.
G.f.: (1-x^4) * Prod_{k>0} (1 + x^(2*k)) * (1 + x^k)^2. - Michael Somos, Feb 10 2005
a(n) ~ 5^(3/4) * Pi * exp(Pi*sqrt(5*n/6)) / (2^(11/4) * 3^(3/4) * n^(5/4)). - Vaclav Kotesovec, Sep 06 2015
EXAMPLE
G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 12*x^4 + 20*x^5 + 32*x^6 + 48*x^7 + 72*x^8 + ...
MATHEMATICA
nmax = 40; CoefficientList[Series[(1-x^4)*Product[(1+x^(2*k))/(1-x^(2*k-1))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 06 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x*O(x^n); polcoeff( (1 - x^4) * eta(x^4 + A) * eta(x^2 + A) / eta(x + A)^2, n))}; /* Michael Somos, Feb 10 2005 */
CROSSREFS
Cf. A080054.
Sequence in context: A300414 A307732 A103258 * A368430 A131770 A322419
KEYWORD
nonn
AUTHOR
Noureddine Chair, Jan 27 2005
EXTENSIONS
Corrected by Vladeta Jovovic, Feb 01 2005
Typo in PARI program fixed by Vaclav Kotesovec, Sep 06 2015
STATUS
approved