A370148 -id:A370148 - 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

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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

A370147 Expansion of ( (1 + x)(1 + 7x)(1 + 49x) )^(1/3).

Original entry on oeis.org

1, 19, -228, 6492, -216372, 7851180, -300848772, 11974587132, -490113592788, 20492868223308, -871404823013412, 37562003034015900, -1637401559515373172, 72053378865932154348, -3196217668534369463748, 142763786831538212246076, -6415201218873454789867284, 289797678008730755585589900

Offset: 0

Paul D. Hanna, Feb 23 2024

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The cube root of F(x) = (1 + x)*(1 + 7*x)*(1 + 49*x) = (1 + 57*x + 399*x^2 + 343*x^3) is an integer series because F(x) == (1+x)^3 (mod 9).

In general, for k > 1, if g.f. = ((1 + x)*(1 + k*x)*(1 + k^2*x))^(1/3), then a(n) ~ (-1)^(n+1) * (k-1)^(2/3) * (k+1)^(1/3) * k^(2*n-1) / (3*Gamma(2/3)*n^(4/3)). - Vaclav Kotesovec, Feb 24 2024

			G.f.: A(x) = 1 + 19*x - 228*x^2 + 6492*x^3 - 216372*x^4 + 7851180*x^5 - 300848772*x^6 + 11974587132*x^7 - 490113592788*x^8 + ...
where A(x)^3 = (1 + 57*x + 399*x^2 + 343*x^3).
RELATED SERIES.
We have the following infinite product involving the g.f. A(x)
A(x)^3 * A(-7*x^2)^3 * A(49*x^3)^3 * A(-343*x^4)^3 * A(2401*x^5)^3 * ... = 1 + 57*x - 19607*x^3 - 47079151*x^6 + 791260232049*x^10 + 93090977300134793*x^15 + ... + (-7)^(n*(n-1)/2) * (1 + (-7)^(2*n+1))/(-6) * x^(n*(n+1)/2) + ...

Paul D. Hanna, Table of n, a(n) for n = 0..406
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Cf. A370145, A370149, A370148.

{a(n) = polcoeff( ( (1 + x)*(1 + 7*x)*(1 + 49*x) +x*O(x^n))^(1/3), n)}
for(n=0, 40, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x)^3 = (1 + x)*(1 + 7*x)*(1 + 49*x) = (1 + 57*x + 399*x^2 + 343*x^3).
(2) Product_{n>=1} A( (-7)^(n-1)*x^n )^3 = Sum_{n>=0} (-7)^(n*(n-1)/2) * (1 + (-7)^(2*n+1))/(-6) * x^(n*(n+1)/2).
a(n) ~ (-1)^(n+1) * 2^(5/3) * 7^(2*n-1) / (3^(1/3) * Gamma(2/3) * n^(4/3)). - Vaclav Kotesovec, Feb 24 2024

A370334 Expansion of [ Sum_{n>=0} 11^(n(n-1)/2) (1 + 11^(2n+1))/12 x^(n*(n+1)/2) ]^(1/3).

Original entry on oeis.org

1, 37, -1369, 133632, -9888768, 845367083, -78838949376, 7721334144755, -776624602305024, 79868229118115328, -8362877755373222400, 888226662691859185152, -95442299152209579505105, 10355840499178710443340288, -1132966823558169033184762368, 124832961812953439236127605357

Offset: 0

Paul D. Hanna, Feb 25 2024

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			G.f.: A(x) = 1 + 37*x - 1369*x^2 + 133632*x^3 - 9888768*x^4 + 845367083*x^5 - 78838949376*x^6 + 7721334144755*x^7 - 776624602305024*x^8 + ...
where the cube of g.f. A(x) yields the series
A(x)^3 = 1 + 111*x + 147631*x^3 + 2161452161*x^6 + 348104014265601*x^10 + 616687495357008127151*x^15 + 12017494675541950940487123311*x^21 + ... + 11^(n*(n-1)/2) * (1 + 11^(2*n+1))/12 * x^(n*(n+1)/2) + ...
The cube of g.f. A(x) also equals the infinite product
A(x)^3 = (1 + x)*(1 - 11*x)*(1 + 11^2*x) * (1 + 11*x^2)*(1 - 11^2*x^2)*(1 + 11^3*x^2) * (1 + 11^2*x^3)*(1 - 11^3*x^3)*(1 + 11^4*x^3) * (1 + 11^3*x^4)*(1 - 11^4*x^4)*(1 + 11^5*x^4) * ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A370149, A370016, A370148, A370019, A370336.

{a(n) = my(A); A = prod(m=1, n+1, (1 + 11^(m-1)*x^m) * (1 - 11^m*x^m) * (1 + 11^(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} 11^(n*(n-1)/2) * (1 + 11^(2*n+1))/12 * x^(n*(n+1)/2).
(2) A(x)^3 = Product_{n>=1} (1 + 11^(n-1)*x^n) * (1 - 11^n*x^n) * (1 + 11^(n+1)*x^n), by the Jacobi triple product identity.
(3) A(x) = Product_{n>=1} F( 11^(n-1)*x^n ), where F(x) = (1 + 111*x - 1221*x^2 - 1331*x^3)^(1/3).
a(n) ~ (-1)^(n+1) * c * 11^(2*n) / n^(4/3), where c = 0.2588865455859866840901787578907966... - Vaclav Kotesovec, Feb 27 2024

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).

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A370147 Expansion of ( (1 + x)(1 + 7x)(1 + 49x) )^(1/3).

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A370334 Expansion of [ Sum_{n>=0} 11^(n(n-1)/2) (1 + 11^(2n+1))/12 x^(n*(n+1)/2) ]^(1/3).

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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).

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A370147 Expansion of ( (1 + x)*(1 + 7*x)*(1 + 49*x) )^(1/3).

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A370334 Expansion of [ Sum_{n>=0} 11^(n*(n-1)/2) * (1 + 11^(2*n+1))/12 * x^(n*(n+1)/2) ]^(1/3).

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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).

A370147 Expansion of ( (1 + x)(1 + 7x)(1 + 49x) )^(1/3).

A370334 Expansion of [ Sum_{n>=0} 11^(n(n-1)/2) (1 + 11^(2n+1))/12 x^(n*(n+1)/2) ]^(1/3).