OFFSET
0,13
COMMENTS
LINKS
Reinhard Zumkeller and Vaclav Kotesovec, Table of n, a(n) for n = 0..20000 (terms 0..300 from Reinhard Zumkeller)
Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.
David J. Hemmer, Generating functions for fixed points of the Mullineux map, arXiv:2402.03643 [math.CO], 2024.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of chi(x) / chi(x^3) in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Sep 20 2013
Expansion of f(x^1, x^5) / f(-x^6) in powers of x where f(,) is a Ramanujan theta function. - Michael Somos, Sep 20 2013
Expansion of G(x^6) * H(-x) + x * G(-x) * H(x^6) where G() (A003114), H() (A003106) are Rogers-Ramanujan functions.
Expansion of q^(-1/12) * eta(q^2)^2 * eta(q^3) * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6)^2) in powers of q.
Euler transform of period 12 sequence [ 1, -1, 0, 0, 1, 0, 1, 0, 0, -1, 1, 0, ...]. - Michael Somos, Jun 26 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227398. - Michael Somos, Sep 20 2013
G.f.: Product_{k>0} (1 - (-x)^k + x^(2*k)).
G.f.: 1 / Product_{k>0} (1 - x^(2*k - 1) + x^(4*k - 2)).
G.f.: 1 / Product_{k>0} ((1 + x^(6*k - 3)) / (1 + x^(2*k - 1))).
G.f.: Product_{k>0} ((1 + x^(6*k - 1)) * (1 + x^(6*k - 5))).
G.f.: 1 / Product_{k>0} (1 + (-x)^(3*k - 1)) * (1 + (-x)^(3*k - 2)).
G.f.: (Sum_{k in Z} x^(k * (3*k - 2))) / (Sum_{k in Z} (-1)^k * x^(3*k * (3*k-1))).
a(n) ~ exp(sqrt(n)*Pi/3)/ (2*sqrt(6)*n^(3/4)) * (1 + (Pi/72 - 9/(8*Pi)) / sqrt(n)). - Vaclav Kotesovec, Aug 30 2015, extended Jan 18 2017
EXAMPLE
E.g. a(25)=5 because 25=19+5+1=17+7+1=13+7+5=13+11+1.
G.f. = 1 + x + x^5 + x^6 + x^7 + x^8 + x^11 + 2*x^12 + 2*x^13 + x^14 + x^16 + ...
G.f. = q + q^13 + q^61 + q^73 + q^85 + q^97 + q^133 + 2*q^145 + 2*q^157 + q^169 + ...
MAPLE
series(product((1+x^(6*k-1))*(1+x^(6*k-5)), k=1..100), x=0, 100);
MATHEMATICA
a[ n_] := SeriesCoefficient[ Product[ 1 - (-x)^k + x^(2 k), {k, n}], {x, 0, n}]; (* Michael Somos, Sep 20 2013 *)
a[ n_] := SeriesCoefficient[ 1 / Product[ 1 - x^k + x^(2 k), {k, 1, n, 2}], {x, 0, n}]; (* Michael Somos, Sep 20 2013 *)
a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, 1, n, 2}] / Product[ 1 + x^k, {k, 3, n, 6}], {x, 0, n}]; (* Michael Somos, Sep 20 2013 *)
a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, 1, n, 6}] Product[ 1 + x^k, {k, 5, n, 6}], {x, 0, n}]; (* Michael Somos, Sep 20 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^6] QPochhammer[ -x^5, x^6], {x, 0, n}]; (* Michael Somos, Sep 20 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] / QPochhammer[ -x^3, x^6], {x, 0, n}]; (* Michael Somos, Sep 20 2013 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2), n))}; /* Michael Somos, Jun 26 2005 */
(PARI) {a(n) = my(A, m); if( n<0, 0, A = x * O(x^n); m = sqrtint(3*n + 1); polcoeff( sum(k= -((m-1)\3), (m+1)\3, x^(k * (3*k - 2)), A) / eta(x^6 + A), n))}; /* Michael Somos, Sep 20 2013 */
(Haskell)
a098884 = p a007310_list where
p _ 0 = 1
p (k:ks) m = if k > m then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Feb 19 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Noureddine Chair, Oct 14 2004
EXTENSIONS
Typo in Maple program fixed by Vaclav Kotesovec, Nov 15 2016
STATUS
approved