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
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
Programs
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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
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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
Comments