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A195848 Expansion of 1 / f(-x^1, -x^5) in powers of x where f() is Ramanujan's two-variable theta function.

%I A195848 #80 Feb 16 2025 08:33:15
%S A195848 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 A195848 94,107,128,153,176,194,213,241,282,331,376,415,456,512,590,680,767,
%U A195848 845,928,1037,1180,1345,1506,1657,1818,2020,2278,2570,2862,3142,3442
%N A195848 Expansion of 1 / f(-x^1, -x^5) in powers of x where f() is Ramanujan's two-variable theta function.
%C A195848 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A195848 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 A195848 The number of partitions of n into parts congruent to 0, 1 or 5 ( mod 6 ). - _Peter Bala_, Dec 09 2020
%H A195848 Seiichi Manyama, <a href="/A195848/b195848.txt">Table of n, a(n) for n = 0..10000</a>
%H A195848 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 A195848 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A195848 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A195848 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 A195848 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 A195848 Euler transform of period 6 sequence [ 1, 0, 0, 0, 1, 1, ...]. - _Michael Somos_, Oct 18 2014
%F A195848 Convolution inverse of A089802. - _Michael Somos_, Oct 18 2014
%F A195848 a(n) ~ exp(Pi*sqrt(n/3))/(4*n). - _Vaclav Kotesovec_, Nov 08 2015
%F A195848 a(n) = (1/n)*Sum_{k=1..n} A284362(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 25 2017
%F A195848 From _Peter Bala_, Dec 09 2020: (Start)
%F A195848 O.g.f.: 1/( Product_{n >= 1} (1 - x^(6*n-5))*(1 - x^(6*n-1))*(1 - x^(6*n)) ).
%F A195848 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 A195848 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 A195848 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 A195848 A001082 := proc(n)
%p A195848         if type(n,'even') then
%p A195848                 n*(3*n-4)/4 ;
%p A195848         else
%p A195848                 (n-1)*(3*n+1)/4 ;
%p A195848         end if;
%p A195848 end proc:
%p A195848 A195838 := proc(n,k)
%p A195848         option remember;
%p A195848         local ks,a,j ;
%p A195848         if A001082(k+1) > n then
%p A195848                 0 ;
%p A195848         elif n <= 5 then
%p A195848                 return 1;
%p A195848         elif k = 1 then
%p A195848                 a := 0 ;
%p A195848                 for j from 1 do
%p A195848                         if A001082(j+1) <= n-1 then
%p A195848                                 a := a+procname(n-1,j) ;
%p A195848                         else
%p A195848                                 break;
%p A195848                         end if;
%p A195848                 end do;
%p A195848                 return a;
%p A195848         else
%p A195848                 ks := A001082(k+1) ;
%p A195848                 (-1)^floor((k-1)/2)*procname(n-ks+1,1) ;
%p A195848         end if;
%p A195848 end proc:
%p A195848 A195848 := proc(n)
%p A195848         A195838(n+1,1) ;
%p A195848 end proc:
%p A195848 seq(A195848(n),n=0..60) ; # _R. J. Mathar_, Oct 07 2011
%t A195848 a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^3] / (QPochhammer[ x] QPochhammer[ x^6]^2), {x, 0, n}]; (* _Michael Somos_, Oct 18 2014 *)
%t A195848 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 A195848 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 A195848 (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 A195848 From _Omar E. Pol_, Jun 10 2012: (Start)
%o A195848 (GW-BASIC)' A program with two A-numbers:
%o A195848 10 Dim A001082(100), A057077(100), a(100): a(0)=1
%o A195848 20 For n = 1 to 58: For j = 1 to n
%o A195848 30 If A001082(j) <= n then a(n) = a(n) + A057077(j-1)*a(n - A001082(j))
%o A195848 40 Next j: Print a(n-1);: Next n (End)
%Y A195848 Column 1 of triangle A195838. Also 1 together with the row sums of triangle A195838. Column 4 of array A195825.
%Y A195848 Cf. A000041, A001082, A006950, A036820, A057077, A195825, A195828, A195849, A195850, A195851, A195852, A196933, A210843, A210964, A211971.
%Y A195848 Cf. A089802.
%K A195848 nonn,easy
%O A195848 0,6
%A A195848 _Omar E. Pol_, Sep 24 2011
%E A195848 New sequence name from _Michael Somos_, Oct 18 2014