A205183 Euler transform of period 5 sequence [ 2, 1, 1, 2, 1, ...].

1, 2, 4, 7, 13, 21, 35, 54, 84, 126, 188, 273, 396, 562, 794, 1106, 1533, 2099, 2862, 3864, 5194, 6927, 9198, 12131, 15938, 20817, 27087, 35070, 45246, 58114, 74400, 94870, 120602, 152768, 192957, 242928, 305025, 381857, 476839, 593830, 737776

Offset: 0

Author

Michael Somos, Jan 23 2012

Keywords

• nonn

Comments

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

Examples

			1 + 2*x + 4*x^2 + 7*x^3 + 13*x^4 + 21*x^5 + 35*x^6 + 54*x^7 + 84*x^8 + ...
q^-7 + 2*q^113 + 4*q^233 + 7*q^353 + 13*q^473 + 21*q^593 + 35*q^713 + ...
		

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. A000041, A003114, A113428.

Programs

Mathematica

f[x_,y_]:= QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; CoefficientList[Series[f[-q^2,-q^3]/QPochhammer[q]^2, {q,0,100}], q] (* G. C. Greubel, Apr 14 2018 *)

PARI

{a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 / (1 - x^k)^[ 1, 2, 1, 1, 2][k%5 + 1], 1 + x * O(x^n)), n))}

Formula

Expansion of G(x) / f(-x) = f(-x^2, -x^3) / f(-x)^2 in powers of x where f() is a Ramanujan theta function and G() is a Rogers-Ramanujan function.

G.f. is the limit as n goes to infinity of Sum_{k=0..n} x^k^2 / ((x;x)k * (x;x){n-k}) = Sum_{k=-n..n} (-1)^k * x^(k*(5*k - 1)/2) / ((x;x){n-k} * (x;x){n+k}).

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

G.f.: (Sum_{k} (-1)^k * x^(k*(5*k - 1)/2)) / Product_{k>0} (1 - x^k)^2.

Convolution of A000041 and A003114.

a(n) ~ exp(Pi*sqrt(14*n/15)) * sqrt(7*phi) / (4*3^(1/2)*5^(3/4)*n), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 24 2018