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Expansion of 1 / f(-x^1, -x^5) in powers of x where f() is Ramanujan's two-variable theta function.
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%I #79 May 17 2024 15:24:59

%S 1,1,1,1,1,2,3,4,4,4,5,7,10,12,13,14,16,21,27,32,35,38,44,54,67,78,86,

%T 94,107,128,153,176,194,213,241,282,331,376,415,456,512,590,680,767,

%U 845,928,1037,1180,1345,1506,1657,1818,2020,2278,2570,2862,3142,3442

%N Expansion of 1 / f(-x^1, -x^5) in powers of x where f() is Ramanujan's two-variable theta function.

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

%C Also column 4 of A195825, therefore this sequence contains two plateaus: [1, 1, 1, 1, 1], [4, 4, 4]. For more information see A210843 and other sequences of this family. - _Omar E. Pol_, Jun 26 2012

%C The number of partitions of n into parts congruent to 0, 1 or 5 ( mod 6 ). - _Peter Bala_, Dec 09 2020

%H Seiichi Manyama, <a href="/A195848/b195848.txt">Table of n, a(n) for n = 0..10000</a>

%H K. Bringmann, J. Lovejoy, and K. Mahlburg, <a href="https://dx.doi.org/10.4310/MRL.2016.v23.n1.a4">A partition identity and the universal mock theta function g_2(x;q)</a>, Mathematical Research Letters, 23 (2016), 67-80.

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of 1 / (psi(x^3) * chi(-x)) in powers of x where psi(), chi() are Ramanujan theta functions. - _Michael Somos_, Jun 07 2012

%F Expansion of q^(1/3) * eta(q^2) * eta(q^3) / (eta(q) * eta(q^6)^2) in powers of q. - _Michael Somos_, Jun 07 2012

%F Euler transform of period 6 sequence [ 1, 0, 0, 0, 1, 1, ...]. - _Michael Somos_, Oct 18 2014

%F Convolution inverse of A089802. - _Michael Somos_, Oct 18 2014

%F a(n) ~ exp(Pi*sqrt(n/3))/(4*n). - _Vaclav Kotesovec_, Nov 08 2015

%F a(n) = (1/n)*Sum_{k=1..n} A284362(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 25 2017

%F From _Peter Bala_, Dec 09 2020: (Start)

%F O.g.f.: 1/( Product_{n >= 1} (1 - x^(6*n-5))*(1 - x^(6*n-1))*(1 - x^(6*n)) ).

%F a(n) = a(n-1) + a(n-5) - a(n-8) - a(n-16) + + - - ... (with the convention a(n) = 0 for negative n), where 1, 5, 8, 16, ... is the sequence of generalized octagonal numbers A001082. (End)

%e G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 4*x^8 + 4*x^9 + 5*x^10 + ...

%e G.f. = 1/q + q^2 + q^5 + q^8 + q^11 + 2*q^14 + 3*q^17 + 4*q^20 + 4*q^23 + 4*q^26 + ...

%p A001082 := proc(n)

%p if type(n,'even') then

%p n*(3*n-4)/4 ;

%p else

%p (n-1)*(3*n+1)/4 ;

%p end if;

%p end proc:

%p A195838 := proc(n,k)

%p option remember;

%p local ks,a,j ;

%p if A001082(k+1) > n then

%p 0 ;

%p elif n <= 5 then

%p return 1;

%p elif k = 1 then

%p a := 0 ;

%p for j from 1 do

%p if A001082(j+1) <= n-1 then

%p a := a+procname(n-1,j) ;

%p else

%p break;

%p end if;

%p end do;

%p return a;

%p else

%p ks := A001082(k+1) ;

%p (-1)^floor((k-1)/2)*procname(n-ks+1,1) ;

%p end if;

%p end proc:

%p A195848 := proc(n)

%p A195838(n+1,1) ;

%p end proc:

%p seq(A195848(n),n=0..60) ; # _R. J. Mathar_, Oct 07 2011

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^3] / (QPochhammer[ x] QPochhammer[ x^6]^2), {x, 0, n}]; (* _Michael Somos_, Oct 18 2014 *)

%t a[ n_] := SeriesCoefficient[ 2 q^(3/8) / (QPochhammer[ q, q^2] EllipticTheta[ 2, 0, q^(3/2)]), {q, 0, n}]; (* _Michael Somos_, Oct 18 2014 *)

%t nmax = 60; CoefficientList[Series[Product[(1+x^k) / ((1+x^(3*k)) * (1-x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 08 2015 *)

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)^2), n))}; /* _Michael Somos_, Jun 07 2012 */

%o From _Omar E. Pol_, Jun 10 2012: (Start)

%o (GW-BASIC)' A program with two A-numbers:

%o 10 Dim A001082(100), A057077(100), a(100): a(0)=1

%o 20 For n = 1 to 58: For j = 1 to n

%o 30 If A001082(j) <= n then a(n) = a(n) + A057077(j-1)*a(n - A001082(j))

%o 40 Next j: Print a(n-1);: Next n (End)

%Y Column 1 of triangle A195838. Also 1 together with the row sums of triangle A195838. Column 4 of array A195825.

%Y Cf. A000041, A001082, A006950, A036820, A057077, A195825, A195828, A195849, A195850, A195851, A195852, A196933, A210843, A210964, A211971.

%Y Cf. A089802.

%K nonn,easy

%O 0,6

%A _Omar E. Pol_, Sep 24 2011

%E New sequence name from _Michael Somos_, Oct 18 2014