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

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, 94, 107, 128, 153, 176, 194, 213, 241, 282, 331, 376, 415, 456, 512, 590, 680, 767, 845, 928, 1037, 1180, 1345, 1506, 1657, 1818, 2020, 2278, 2570, 2862, 3142, 3442

Offset: 0

Author

Omar E. Pol, Sep 24 2011

Keywords

• nonn
• easy

Comments

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

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

The number of partitions of n into parts congruent to 0, 1 or 5 ( mod 6 ). - Peter Bala, Dec 09 2020

Examples

			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 + ...
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 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• K. Bringmann, J. Lovejoy, and K. Mahlburg, A partition identity and the universal mock theta function g_2(x;q), Mathematical Research Letters, 23 (2016), 67-80.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

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

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

Cf. A089802.

Programs

Maple

A001082 := proc(n)
        if type(n,'even') then
                n*(3*n-4)/4 ;
        else
                (n-1)*(3*n+1)/4 ;
        end if;
end proc:
A195838 := proc(n,k)
        option remember;
        local ks,a,j ;
        if A001082(k+1) > n then
                0 ;
        elif n <= 5 then
                return 1;
        elif k = 1 then
                a := 0 ;
                for j from 1 do
                        if A001082(j+1) <= n-1 then
                                a := a+procname(n-1,j) ;
                        else
                                break;
                        end if;
                end do;
                return a;
        else
                ks := A001082(k+1) ;
                (-1)^floor((k-1)/2)*procname(n-ks+1,1) ;
        end if;
end proc:
A195848 := proc(n)
        A195838(n+1,1) ;
end proc:
seq(A195848(n),n=0..60) ; # R. J. Mathar, Oct 07 2011

Mathematica

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

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 */
From Omar E. Pol, Jun 10 2012: (Start)
(GW-BASIC)' A program with two A-numbers:
10 Dim A001082(100), A057077(100), a(100): a(0)=1
20 For n = 1 to 58: For j = 1 to n
30 If A001082(j) <= n then a(n) = a(n) + A057077(j-1)*a(n - A001082(j))
40 Next j: Print a(n-1);: Next n (End)

Formula

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

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

Euler transform of period 6 sequence [ 1, 0, 0, 0, 1, 1, ...]. - Michael Somos, Oct 18 2014

Convolution inverse of A089802. - Michael Somos, Oct 18 2014

a(n) ~ exp(Pi*sqrt(n/3))/(4*n). - Vaclav Kotesovec, Nov 08 2015

a(n) = (1/n)*Sum_{k=1..n} A284362(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

From Peter Bala, Dec 09 2020: (Start)

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

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)

Extensions

New sequence name from Michael Somos, Oct 18 2014