A370337 Expansion of Product_{n>=1} (1 - 2^(n-1)*x^n) * (1 + 2^(n-1)*x^n)^2.
1, 1, 1, 5, 6, 22, 40, 108, 192, 536, 1072, 2528, 5344, 12288, 26624, 61312, 129024, 286720, 646656, 1389568, 3028992, 6717440, 14708736, 31604736, 69763072, 150110208, 329809920, 714473472, 1546649600, 3324772352, 7332954112, 15626403840, 33840693248, 73194799104, 158456610816
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + x^2 + 5*x^3 + 6*x^4 + 22*x^5 + 40*x^6 + 108*x^7 + 192*x^8 + 536*x^9 + 1072*x^10 + 2528*x^11 + 5344*x^12 + ... where A(x) is the series expansion of the infinite product given by A(x) = (1 - x)*(1 + x)^2 * (1 - 2*x^2)*(1 + 2*x^2)^2 * (1 - 4*x^3)*(1 + 4*x^3)^2 * (1 - 8*x^4)*(1 + 8*x^4)^2 * ... * (1 - 2^(n-1)*x^n)*(1 + 2^(n-1)*x^n)^2 * ... Compare A(x) to the series that results from a similar infinite product: (1 - 2*x)*(1 + 2*x)^2 * (1 - 4*x^2)*(1 + 4*x^2)^2 * (1 - 8*x^3)*(1 + 8*x^3)^2 * (1 - 16*x^4)*(1 + 16*x^4)^2 * ... = 1 + 2*x + 8*x^3 + 64*x^6 + 1024*x^10 + 32768*x^15 + 2097152*x^21 + 268435456*x^28 + 68719476736*x^36 + ...
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..1035
Programs
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Mathematica
CoefficientList[Series[8*QPochhammer[1/2, 2*x] * QPochhammer[-1/2, 2*x]^2/9, {x, 0, 40}], x] (* Vaclav Kotesovec, Feb 26 2024 *) -
PARI
{a(n) = polcoeff( prod(k=1,n, (1 - 2^(k-1)*x^k) * (1 + 2^(k-1)*x^k)^2 +x*O(x^n)), n)} for(n=0,40, print1(a(n),", "))
Formula
a(n) ~ c^(1/4) * 2^n * exp(sqrt(c*n)) / (3*sqrt(Pi)*n^(3/4)), where c = 2*log(2)^2 - Pi^2/3 - 8*polylog(2,-1/2) = Pi^2 + 6*log(2)^2 + 8*polylog(2,-2) = 1.258351549529119595933889966687474131697... - Vaclav Kotesovec, Feb 26 2024
Comments