A370765 -id:A370765 - cp's OEIS frontend

A370016 Expansion of A(x) = [ Sum_{n>=0} 2^(n(n-1)/2) (1 + 2^(2n+1))/3 x^(n*(n+1)/2) ]^(1/3).

1, 1, -1, 9, -18, 44, -54, 350, -1359, 3789, -9585, 42489, -163900, 543474, -1933092, 7499404, -27668718, 100329714, -371138346, 1394575578, -5236658316, 19587163968, -73536845444, 278088068628, -1052804678958, 3985553554074, -15132118280498, 57617112474306, -219680808219216

Offset: 0

Paul D. Hanna, Feb 22 2024

sign

Equals the self-convolution cube root of A370015.

			G.f.: A(x) = 1 + x - x^2 + 9*x^3 - 18*x^4 + 44*x^5 - 54*x^6 + 350*x^7 - 1359*x^8 + 3789*x^9 - 9585*x^10 + 42489*x^11 - 163900*x^12 + 543474*x^13 - 1933092*x^14 + 7499404*x^15 + ...
where the cube of g.f. A(x) yields the series
A(x)^3 = 1 + 3*x + 22*x^3 + 344*x^6 + 10944*x^10 + 699392*x^15 + 89489408*x^21 + 22907191296*x^28 + 11728213508096*x^36 + ... + 2^(n*(n-1)/2)*(1 + 2^(2*n+1))/3 * x^(n*(n+1)/2) + ...
The cube of g.f. A(x) also equals the infinite product
A(x)^3 = (1 + x)*(1 - 2*x)*(1 + 2^2*x) * (1 + 2*x^2)*(1 - 2^2*x^2)*(1 + 2^3*x^2) * (1 + 2^2*x^3)*(1 - 2^3*x^3)*(1 + 2^4*x^3) * (1 + 2^3*x^4)*(1 - 2^4*x^4)*(1 + 2^5*x^4) * ...
Equivalently,
A(x) = (1 + 3*x - 6*x^2 - 8*x^3)^(1/3) * (1 + 6*x^2 - 24*x^4 - 64*x^6)^(1/3) * (1 + 12*x^3 - 96*x^6 - 512*x^9)^(1/3) * (1 + 24*x^4 - 384*x^8 - 4096*x^12)^(1/3) * (1 + 48*x^5 - 1536*x^10 - 32768*x^15)^(1/3) * ...
where
(1 + 3*x - 6*x^2 - 8*x^3)^(1/3) = 1 + x - 3*x^2 + 3*x^3 - 12*x^4 + 30*x^5 - 102*x^6 + 318*x^7 - 1083*x^8 + 3657*x^9 + ... + A370145(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..630

Cf. A370015, A370145, A370148, A370334, A370019, A370336.

Cf. A304961, A370716, A370765.

Round[CoefficientList[Series[QPochhammer[-2, 2*x]^(1/3) * QPochhammer[-1/2, 2*x]^(1/3) * QPochhammer[2*x]^(1/3)*2^(1/3)/3^(2/3), {x, 0, 30}], x]] (* Vaclav Kotesovec, Feb 23 2024 *)

{a(n) = my(A);
A = prod(m=1, n+1, (1 + 2^(m-1)*x^m) * (1 - 2^m*x^m) * (1 + 2^(m+1)*x^m) +x*O(x^n))^(1/3);
polcoeff(H=A, n)}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x)^3 = Sum_{n>=0} 2^(n*(n-1)/2) * (1 + 2^(2*n+1))/3 * x^(n*(n+1)/2).
(2) A(x)^3 = Product_{n>=1} (1 + 2^(n-1)*x^n) * (1 - 2^n*x^n) * (1 + 2^(n+1)*x^n), by the Jacobi triple product identity.
(3) A(x) = Product_{n>=1} F( 2^(n-1)*x^n ), where F(x) = (1 + 3*x - 6*x^2 - 8*x^3)^(1/3).
a(n) ~ c * (-1)^(n+1) * 4^n / n^(4/3), where c = 0.260357663494676514371316... - Vaclav Kotesovec, Feb 23 2024
Radius of convergence r = 1/4 (from Vaclav Kotesovec's formula) and A(r) = ( Sum_{n>=0} (1/2^n + 2*2^n)/3 * 1/2^(n*(n+1)/2) )^(1/3) = ( Product_{n>=1} (1 + 1/2^n)*(1 - 1/4^(n+1)) )^(1/3) = 1.298389210904220681888354941631161162163... - Paul D. Hanna, Mar 07 2024

A370761 Expansion of Product_{k>=1} (1 + 2^(k+1)x^k) (1 + 2^(k-1)*x^k).

Original entry on oeis.org

1, 5, 14, 70, 196, 640, 2248, 6480, 19072, 56000, 169792, 466560, 1327104, 3642880, 10030080, 27776000, 74541056, 199065600, 531505152, 1401405440, 3672801280, 9674588160, 25018564608, 64701071360, 166363136000, 426159636480, 1084287352832, 2756737761280, 6979072294912

Offset: 0

Vaclav Kotesovec, Mar 01 2024

nonn

Cf. A032302, A304961, A370016, A370764, A370765, A370792.

nmax = 30; CoefficientList[Series[Product[(1 + 2^(k+1)*x^k)*(1 + 2^(k-1)*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ (Pi^2/3 + log(2)^2)^(1/4) * 2^(n - 3/4) * exp(sqrt(2*(Pi^2/3 + log(2)^2)*n)) / (3*sqrt(Pi)*n^(3/4)).

A370764 a(n) = 4^n * [x^n] Product_{k>=1} ((1 + 2^(k+1)x^k) (1 + 2^(k-1)*x^k))^(1/2).

Original entry on oeis.org

1, 10, 62, 1620, 6966, 157580, 1284012, 19189160, 73908774, 2233414620, 9656822916, 287668788120, -324007115716, 40151699854200, -199460032590312, 7130611518222160, -64971542557275642, 1292318115470489340, -15433712240157937260, 265667290368470451000, -3624776372747687578668

Offset: 0

Vaclav Kotesovec, Mar 01 2024

sign

In general, if d > 1 and g.f. = Product_{k>=1} ((1 + d^(k+1)*x^k) * (1 + d^(k-1)*x^k))^(1/2), then a(n) ~ (-1)^(n+1) * QPochhammer(-1/d) * d^(2*n) / (2*sqrt((1 + 1/d)*Pi) * n^(3/2)).

Cf. A032302, A304961, A370709, A370761, A370765.

nmax = 25; CoefficientList[Series[Product[(1+2^(k+1)*x^k)*(1+2^(k-1)*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x] * 4^Range[0, nmax]
nmax = 25; CoefficientList[Series[Product[(1+2^(3*k+1)*x^k)*(1+2^(3*k-1)*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x]
nmax = 25; CoefficientList[Series[(2*QPochhammer[-2, x]*QPochhammer[-1/2, x])^(1/2)/3, {x, 0, nmax}], x] * 8^Range[0, nmax]

G.f.: Product_{k>=1} ((1 + 2^(3*k+1)*x^k) * (1 + 2^(3*k-1)*x^k))^(1/2).

a(n) ~ (-1)^(n+1) * c * 16^n / n^(3/2), where c = QPochhammer(-1/2) / sqrt(6*Pi) = 0.278865402428524528968820654198674...

cp's OEIS Frontend

A370016 Expansion of A(x) = [ Sum_{n>=0} 2^(n(n-1)/2) (1 + 2^(2n+1))/3 x^(n*(n+1)/2) ]^(1/3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A370761 Expansion of Product_{k>=1} (1 + 2^(k+1)x^k) (1 + 2^(k-1)*x^k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A370764 a(n) = 4^n * [x^n] Product_{k>=1} ((1 + 2^(k+1)x^k) (1 + 2^(k-1)*x^k))^(1/2).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

A370016 Expansion of A(x) = [ Sum_{n>=0} 2^(n*(n-1)/2) * (1 + 2^(2*n+1))/3 * x^(n*(n+1)/2) ]^(1/3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A370761 Expansion of Product_{k>=1} (1 + 2^(k+1)*x^k) * (1 + 2^(k-1)*x^k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A370764 a(n) = 4^n * [x^n] Product_{k>=1} ((1 + 2^(k+1)*x^k) * (1 + 2^(k-1)*x^k))^(1/2).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A370016 Expansion of A(x) = [ Sum_{n>=0} 2^(n(n-1)/2) (1 + 2^(2n+1))/3 x^(n*(n+1)/2) ]^(1/3).

A370761 Expansion of Product_{k>=1} (1 + 2^(k+1)x^k) (1 + 2^(k-1)*x^k).

A370764 a(n) = 4^n * [x^n] Product_{k>=1} ((1 + 2^(k+1)x^k) (1 + 2^(k-1)*x^k))^(1/2).