login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A098884 Number of partitions of n into distinct parts in which each part is congruent to 1 or 5 mod 6. 10
1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 2, 2, 1, 0, 1, 2, 3, 3, 2, 1, 1, 3, 5, 5, 3, 1, 2, 5, 7, 7, 5, 3, 3, 7, 11, 11, 7, 4, 6, 11, 15, 15, 11, 7, 8, 15, 22, 22, 15, 10, 13, 22, 30, 30, 23, 16, 18, 30, 42, 42, 31, 22, 27, 43, 56, 56, 44, 33, 37, 57, 77, 77, 59, 45, 53, 79, 101, 101, 82, 64, 71 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,13

COMMENTS

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

Convolution of A281244 and A280456. - Vaclav Kotesovec, Jan 18 2017

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.

M. 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))).

A109389(n) = (-1)^n * a(n). Convolution inverse of A227398.

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

Cf. A109389, A227398.

Cf. A007310, A056970, A096981, A097451.

Sequence in context: A180835 A053188 A109389 * A297964 A296239 A039800

Adjacent sequences:  A098881 A098882 A098883 * A098885 A098886 A098887

KEYWORD

nonn

AUTHOR

Noureddine Chair, Oct 14 2004

EXTENSIONS

Typo in Maple program fixed by Vaclav Kotesovec, Nov 15 2016

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 19 12:48 EST 2018. Contains 299333 sequences. (Running on oeis4.)