A098613 Expansion of psi(x^2) / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.

1, 1, 3, 4, 7, 10, 17, 23, 35, 48, 69, 93, 131, 173, 236, 310, 413, 536, 704, 903, 1170, 1489, 1904, 2403, 3044, 3811, 4784, 5951, 7409, 9157, 11325, 13912, 17095, 20891, 25519, 31029, 37708, 45632, 55184, 66495, 80050, 96064, 115173, 137680, 164425, 195860

Offset: 0

Author

Michael Somos, Sep 17 2004

Keywords

• nonn

Comments

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

This sequence convolved with A000009 gives A001936. - Gary W. Adamson, Mar 24 2011

a(n) is the number of partitions of n in which each odd part can occur any number of times but each even part is of two kinds and each kind can occur at most once. - Michael Somos, Dec 01 2019

Examples

			G.f. = 1 + x + 3*x^2 + 4*x^3 + 7*x^4 + 10*x^5 + 17*x^6 + 23*x^7 + ...
G.f. = q^5 + q^29 + 3*q^53 + 4*q^77 + 7*q^101 + 10*q^125 + 17*q^149 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A029552, A083906, A143161.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] / (2 x^(1/4) QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, Oct 29 2013 *)
a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x, x^2] QPochhammer[ x^2, x^4]^2), {x, 0, n}]; (* Michael Somos, Oct 29 2013 *)
nmax = 40; CoefficientList[ Series[Product[(1 + x^k) * (1 + x^(2*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^2, x^2]^3, {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x]^2 QPochhammer[ -x, x]^3, {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ -x^2, x^2]^2, {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)

PARI

{a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(4*n+1)-1)\2, x^(k^2+k)) / eta(x + x * O(x^n)), n))};

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 / (eta(x + A) * eta(x^2 + A)), n))};

PARI

{a(n) = if( n<0,0, polcoeff( sum(k=0, 2*n+1, prod(i=1, k, (1 - x^(2*n+2 - i)) / (1 - x^i))) / 2, n^2))};

Formula

Expansion of chi(x) / chi(-x^2)^3 = 1 / (chi(-x)* chi(-x^2)^2) = 1 / (chi(x)^2 * chi(-x)^3) in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Sep 07 2015

Expansion of q^(-5/24) * eta(q^4)^2 / (eta(q) * eta(q^2)) in powers of q.

Euler transform of period 4 sequence [1, 2, 1, 0, ...].

G.f. A(x) is the limit of x^(n^2+n) * P_{2*n+1}(1/x)/2 where P_n(q) = Sum_{k=0..n} C(n, k; q) and C(n, k; q) is the q-binomial coefficients. See A083906 for P_n(q).

G.f.: (Sum_{k>0} x^(k^2-k)) / (Product_{k>0} (1 - x^k)).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(11/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015

G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 8^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A143161. - Michael Somos, Sep 07 2015

G.f.: Product_{k>=1} (1 + x^(2*k))^2 / (1 - x^(2*k-1)). - Michael Somos, Dec 01 2019