OFFSET
0,5
COMMENTS
Number 39 of the 130 identities listed in Slater 1952.
Number of partitions of 2*n into distinct odd parts. - Vladeta Jovovic, May 08 2003
REFERENCES
M. D. Hirschhorn, The Power of q, Springer, 2017. Chapter 19, Exercises p. 173.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
G. E. Andrews et al., q-Engel series expansions and Slater's identities, Quaestiones Math., 24 (2001), 403-416.
George E. Andrews, Jethro van Ekeren and Reimundo Heluani, The singular support of the Ising model, arXiv:2005.10769 [math.QA], 2020. See (1.4.2) p. 2.
T. Gannon, G. Hoehn, H. Yamauchi, et. al., VOA Unitary Minimal Model m=1, character.
M. D. Hirschhorn, Some partition theorems of the Rogers-Ramanujan type, J. Combin. Theory Ser. A 27 (1979), no. 1, 33-37. MR0541341 (80j:05010). See Theorem 4. [From N. J. A. Sloane, Mar 19 2012]
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
Lucy Joan Slater, Further Identities of the Rogers-Ramanujan Type, Proc. London Math. Soc., Series 2, vol.s2-54, no.2, pp. 147-167, (1952).
Eric Weisstein's World of Mathematics, Jackson-Slater Identity
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
M. P. Zaletel and R. S. K. Mong, Exact Matrix Product States for Quantum Hall Wave Functions, arXiv preprint arXiv:1208.4862 [cond-mat.str-el], 2012. - N. J. A. Sloane, Dec 25 2012
FORMULA
Euler transform of period 16 sequence [0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, ...]. - Michael Somos, Apr 11 2004
G.f.: Sum_{n>=0} q^(2*n^2) / Product_{k=1..2*n} (1 - q^k). - Joerg Arndt, Apr 01 2014
a(n) ~ exp(sqrt(n/3)*Pi) / (2^(5/2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 04 2015
Expansion of f(x^3, x^5) / f(-x^2) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Apr 14 2016
a(n) = A000700(2*n).
a(n) = A027356(4n+1,2n+1). - Alois P. Heinz, Oct 28 2019
From Peter Bala, Feb 08 2021: (Start)
G.f.: A(x) = Product_{n >= 1} (1 + x^(4*n))^2*(1 + x^(4*n-2))*(1 + x^(8*n-3))*(1 + x^(8*n-5)).
The 2 X 2 matrix Product_{k >= 0} [1, x^(2*k+1); x^(2*k+1), 1] = [A(x^2), x*B(x^2); x*B(x)^2, A(x^2)], where B(x) is the g.f. of A069911.
A(x^2) + x*B(x^2) = A^2(-x) + x*B^2(-x) = Product_{k >= 0} 1 + x^(2*k+1), the g.f. of A000700.
A^2(x) + x*B^2(x) is the g.f. of A226622.
(A^2(x) + x*B^2(x))/(A^2(x) - x*B^2(x)) is the g.f. of A208850.
A^4(sqrt(x)) - x*B^4(sqrt(x)) is the g.f. of A029552.
EXAMPLE
G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 5*x^8 + 5*x^9 + ...
G.f. = q^-1 + q^95 + q^143 + 2*q^191 + 2*q^239 + 3*q^287 + 3*q^335 + ...
MAPLE
a:= proc(n) option remember; `if`(n=0, 1,
add(add(d*[0$2, 1$4, 0$5, 1$4, 0][irem(d, 16)+1],
d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..80); # Alois P. Heinz, Apr 01 2014
MATHEMATICA
max = 56; p = Product[1/(1-x^i), {i, Select[Range[max], MemberQ[{2, 3, 4, 5, 11, 12, 13, 14}, Mod[#, 16]]&]}]; s = Series[p, {x, 0, max}]; a[n_] := Coefficient[s, x, n]; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Apr 09 2014 *)
nmax=60; CoefficientList[Series[Product[(1-x^(8*k-1))*(1-x^(8*k-7))*(1-x^(8*k))*(1-x^(16*k-6))*(1-x^(16*k-10))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 04 2015 *)
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^-{ 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0 }[[ Mod[k, 16] + 1]], {k, n}], {x, 0, n}]; (* Michael Somos, Apr 14 2016 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, n=2*n; A = x * O(x^n); polcoeff( eta(-x + A) / eta(x^2 + A), n))}; /* Michael Somos, Apr 11 2004 */
(PARI) N=66; q='q+O('q^N); S=1+sqrtint(N);
gf=sum(n=0, S, q^(2*n^2) / prod(k=1, 2*n, 1-q^k ) );
Vec(gf) \\ Joerg Arndt, Apr 01 2014
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^-[ 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0][k%16 + 1]), n))}; /* Michael Somos, Apr 14 2016 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, May 05 2002
STATUS
approved