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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

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 * A131770 A322419 A246850

Adjacent sequences:  A100681 A100682 A100683 * A100685 A100686 A100687

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

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Last modified June 24 18:46 EDT 2021. Contains 345419 sequences. (Running on oeis4.)