A370016 -id:A370016 - cp's OEIS frontend

A370145 Expansion of ( (1 + x)(1 - 2x)(1 + 4x) )^(1/3).

1, 1, -3, 3, -12, 30, -102, 318, -1083, 3657, -12747, 44715, -159222, 571332, -2068608, 7538664, -27646374, 101915850, -377496030, 1404077790, -5242135728, 19637862132, -73793090676, 278068062756, -1050503580534, 3977985415746, -15096209345958, 57403753019238, -218683959367908

Offset: 0

Paul D. Hanna, Feb 23 2024

sign

The cube root of F(x) = (1 + x)*(1 - 2*x)*(1 + 4*x) = (1 + 3*x - 6*x^2 - 8*x^3) is an integer series because F(x) == (1+x)^3 (mod 9).

			G.f.: A(x) = 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 - 12747*x^10 + 44715*x^11 - 159222*x^12 + ...
where A(x)^3 = (1 + 3*x - 6*x^2 - 8*x^3).
RELATED SERIES.
The following infinite product equals the g.f. of A370015:
A(x)^3 * A(2*x^2)^3 * A(4*x^3)^3 * A(8*x^4)^3 * ... * A(2^(n-1)*x^n)^3 * ... = 1 + 3*x + 22*x^3 + 344*x^6 + 10944*x^10 + 699392*x^15 + 89489408*x^21 + 22907191296*x^28 + ... + 2^(n*(n-1)/2)*(1 + 2^(2*n+1))/3 * x^(n*(n+1)/2) + ... by the Jacobi triple product identity.
If A(x) = 1/B(x/A(x)) then B(x) = 1/A(x/B(x)) begins
B(x) = 1 - x + 3*x^2 - 9*x^6 + 27*x^8 - 324*x^12 + 1215*x^14 - 18711*x^18 + 75816*x^20 - 1301265*x^24 + 5484996*x^26 - 100048689*x^30 + 431943435*x^32 - 8192222064*x^36 + ... + A370146(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..630
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. A370015, A370016, A370146, A370147, A370149.

CoefficientList[Series[Surd[(1+x)(1-2x)(1+4x),3],{x,0,30}],x] (* Harvey P. Dale, Oct 04 2024 *)

{a(n) = polcoeff( (1 + 3*x - 6*x^2 - 8*x^3 +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 - 2*x)*(1 + 4*x) = (1 + 3*x - 6*x^2 - 8*x^3).
(2) Product_{n>=1} A( 2^(n-1)*x^n )^3 = Sum_{n>=0} 2^(n*(n-1)/2)*(1 + 2^(2*n+1))/3 * x^(n*(n+1)/2), which is the g.f. of A370015.
(3) A(x) = 1/B(x/A(x)) where B(x) = 1/A(x/B(x)) = x/Series_Reversion(x/A(x)) equals the g.f. of A370146.
a(n) ~ (-1)^(n+1) * 2^(2*n-1) / (3^(1/3) * Gamma(2/3) * n^(4/3)). - Vaclav Kotesovec, Feb 23 2024

A370019 Expansion of [ Sum_{n>=0} (-4)^n * (24^n + 1)/3 x^(n*(n+1)/2) ]^(1/3).

Original entry on oeis.org

1, -4, -16, -48, -384, -2816, -24384, -206336, -1815552, -16189440, -146777856, -1346648064, -12487131136, -116810932224, -1101080592384, -10447586845696, -99706199973888, -956400813293568, -9215587975397376, -89158545637244928, -865730439117078528, -8433936444598677504

Offset: 0

Paul D. Hanna, Feb 23 2024

sign

The self-convolution cube equals A370018.

			G.f.: A(x) = 1 - 4*x - 16*x^2 - 48*x^3 - 384*x^4 - 2816*x^5 - 24384*x^6 - 206336*x^7 - 1815552*x^8 - 16189440*x^9 - 146777856*x^10 + ...
RELATED SERIES.
The cube of the g.f. A(x) equals the g.f. A370018 which starts as
A(x)^3 = 1 - 12*x + 176*x^3 - 2752*x^6 + 43776*x^10 - 699392*x^15 + 11186176*x^21 + ... + (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2) + ...
The reciprocal of the g.f. A(x) equals the g.f. of A370044, which begins
1/A(x) = 1 + 4*x + 32*x^2 + 240*x^3 + 2048*x^4 + 17920*x^5 + 163904*x^6 + 1526784*x^7 + 14473216*x^8 + 138743808*x^9 + ... + A370044(n)*x^n + ...

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

Cf. A370016, A370044, A370336.

{a(n) = my(A);
A = sum(m=0, sqrtint(2*n+1), (-4)^m * (1 + 2*4^m)/3 * x^(m*(m+1)/2) +x*O(x^n))^(1/3);
polcoeff(H=A, n)}
for(n=0, 20, print1(a(n), ", "))

a(n) ~ c * d^n / n^(4/3), where d = 10.39336299855957350315151176284030870108168399888... and c = -0.218294054014127126766352511836393819909572679... - Vaclav Kotesovec, Feb 24 2024

A370336 Expansion of [ Sum_{n>=0} 5^n * (24^n + 1)/3 x^(n*(n+1)/2) ]^(1/3).

Original entry on oeis.org

1, 5, -25, 300, -3000, 34375, -426750, 5539375, -73968750, 1010175000, -14043011250, 198006675000, -2824523453125, 40684553625000, -590871274218750, 8642318714253125, -127185323309250000, 1881843237600000000, -27976771190059687500, 417688301999460937500, -6259735680122821875000

Offset: 0

Paul D. Hanna, Feb 23 2024

sign

Self-convolution cube equals A370335.

			G.f.: A(x) = 1 + 5*x - 25*x^2 + 300*x^3 - 3000*x^4 + 34375*x^5 - 426750*x^6 + 5539375*x^7 - 73968750*x^8 + 1010175000*x^9 - 14043011250*x^10 + ...
RELATED SERIES.
The cube of the g.f. A(x) yields the g.f. of A370335 starting as
A(x)^3 = 1 + 15*x + 275*x^3 + 5375*x^6 + 106875*x^10 + 2134375*x^15 + 42671875*x^21 + 853359375*x^28 + 17066796875*x^36 + ... + 5^n*(2*4^n + 1)/3 * x^(n*(n+1)/2) + ...

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

Cf. A370335, A370016.

{a(n) = my(A);
A = sum(m=0, sqrtint(2*n+1), 5^m*(2*4^m + 1)/3 * x^(m*(m+1)/2) +x*O(x^n))^(1/3);
polcoeff(H=A, n)}
for(n=0, 20, print1(a(n), ", "))

a(n) ~ (-1)^(n+1) * c * d^n / n^(4/3), where d = 16.061038491618401040959460250524051290971925631740259277535... and c = 0.25648790376068702946627569573532916303248367815529074... - Vaclav Kotesovec, Feb 24 2024

A370015 Expansion of A(x) = Sum_{n>=0} 2^(n(n-1)/2) (1 + 2^(2n+1))/3 x^(n*(n+1)/2).

Original entry on oeis.org

1, 3, 0, 22, 0, 0, 344, 0, 0, 0, 10944, 0, 0, 0, 0, 699392, 0, 0, 0, 0, 0, 89489408, 0, 0, 0, 0, 0, 0, 22907191296, 0, 0, 0, 0, 0, 0, 0, 11728213508096, 0, 0, 0, 0, 0, 0, 0, 0, 12009621912813568, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24595670493070098432, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 100743830310818104213504

Offset: 0

Paul D. Hanna, Feb 22 2024

nonn

Equals the self-convolution cube of A370016.

			G.f.: A(x) = 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 g.f. A(x) equals the infinite product
A(x) = (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 + 6*x^2 - 24*x^4 - 64*x^6) * (1 + 12*x^3 - 96*x^6 - 512*x^9) * (1 + 24*x^4 - 384*x^8 - 4096*x^12 ) * (1 + 48*x^5 - 1536*x^10 - 32768*x^15) * ...
Notice that the cube root of A(x) yields an integer series
A(x)^(1/3) = 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 + ... + A370016(n)*x^n + ...

Cf. A370016.

{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));
polcoeff(H=A, n)}
for(n=0, 66, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = Sum_{n>=0} 2^(n*(n-1)/2) * (1 + 2^(2*n+1))/3 * x^(n*(n+1)/2).
(2) A(x) = 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).

A370148 Expansion of A(x) = [ Sum_{n>=0} (-7)^(n(n-1)/2) (1 + (-7)^(2n+1))/(-6) x^(n*(n+1)/2) ]^(1/3).

Original entry on oeis.org

1, 19, -361, 4896, -186048, 6361181, -265706784, 10569322565, -439680983904, 18480280546656, -790074277452000, 34174424338394976, -1494143747622128305, 65898152303725266336, -2928713377590693411552, 131019840536990930329051, -5895300394280706457304448, 266614701826937350737301056

Offset: 0

Paul D. Hanna, Feb 23 2024

sign

			G.f.: A(x) = 1 + 19*x - 361*x^2 + 4896*x^3 - 186048*x^4 + 6361181*x^5 - 265706784*x^6 + 10569322565*x^7 - 439680983904*x^8 + 18480280546656*x^9 + ...
The cube of g.f. A(x) equals the infinite product
A(x)^3 = (1 + x)*(1 + 7*x)*(1 + 7^2*x) * (1 - 7*x^2)*(1 - 7^2*x^2)*(1 - 7^3*x^2) * (1 + 7^2*x^3)*(1 + 7^3*x^3)*(1 + 7^4*x^3) * (1 - 7^3*x^4)*(1 - 7^4*x^4)*(1 - 7^5*x^4) * ...
Notice that the cube of A(x) yields the series
A(x)^3 = 1 + 57*x - 19607*x^3 - 47079151*x^6 + 791260232049*x^10 + 93090977300134793*x^15 - 76664422756665399911143*x^21 + ... + (-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

Cf. A370147, A370016, A370334, A370019, A370336.

{a(n) = my(A);
A = prod(m=1, n+1, (1 + (-7)^(m-1)*x^m) * (1 - (-7)^m*x^m) * (1 + (-7)^(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} (-7)^(n*(n-1)/2) * (1 + (-7)^(2*n+1))/(-6) * x^(n*(n+1)/2).
(2) A(x)^3 = Product_{n>=1} (1 + (-7)^(n-1)*x^n) * (1 - (-7)^n*x^n) * (1 + (-7)^(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 + 57*x + 399*x^2 + 343*x^3)^(1/3) which is the g.f. of A370147.
a(n) ~ (-1)^(n+1) * c * 7^(2*n) / n^(4/3), where c = 0.2168488573077459727164856825904737112... - 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

sign

			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

A370146 Expansion of x / Series_Reversion( x/(1 + 3x - 6x^2 - 8*x^3)^(1/3) ).

Original entry on oeis.org

1, -1, 3, 0, 0, 0, -9, 0, 27, 0, 0, 0, -324, 0, 1215, 0, 0, 0, -18711, 0, 75816, 0, 0, 0, -1301265, 0, 5484996, 0, 0, 0, -100048689, 0, 431943435, 0, 0, 0, -8192222064, 0, 35942240565, 0, 0, 0, -700434986472, 0, 3108770417700, 0, 0, 0, -61805774132388, 0, 276711654879477

Offset: 0

Paul D. Hanna, Feb 23 2024

sign

The cube root of F(x) = (1 + 3*x - 6*x^2 - 8*x^3) = (1 + x)*(1 - 2*x)*(1 + 4*x) is an integer series because F(x) == (1+x)^3 (mod 9).

			G.f.: A(x) = = 1 - x + 3*x^2 - 9*x^6 + 27*x^8 - 324*x^12 + 1215*x^14 - 18711*x^18 + 75816*x^20 - 1301265*x^24 + 5484996*x^26 - 100048689*x^30 + ...
RELATED SERIES.
If A(x) = 1/B(x/A(x)) then B(x) = (1 + 3*x - 6*x^2 - 8*x^3)^(1/3) begins
B(x) = 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 - 12747*x^10 + 44715*x^11 + ... + A370145(n)*x^n + ...
A(x) = 1/D(x^6) + 3*x^2*D(x^6) - x, where
1/D(x) = 1 - 9*x - 324*x^2 - 18711*x^3 - 1301265*x^4 - 100048689*x^5 - 8192222064*x^6 - 700434986472*x^7 + ...
and D(x) = ( (1-sqrt(1-108*x))/(54*x) )^(1/3) begins
D(x) = 1 + 9*x + 405*x^2 + 25272*x^3 + 1828332*x^4 + 143981145*x^5 + 11980746855*x^6 + 1036256805900*x^7 + ... + 3^n*A008931(n)*x^n + ...

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

Cf. A370016, A370145, A008931, A000108.

{a(n) = polcoeff( x/serreverse( x/(1 + 3*x - 6*x^2 - 8*x^3 +x*O(x^n))^(1/3) ), n)}
for(n=0,50,print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = x / Series_Reversion( x/(1 + 3*x - 6*x^2 - 8*x^3)^(1/3) ).
(2) A(x) = 1 / B(x/A(x)) where B(x) = 1 / A(x/B(x)) = (1 + 3*x - 6*x^2 - 8*x^3)^(1/3) equals the g.f. of A370145.
(3) A(x) = 1/C(27*x^6)^(1/3) + 3*x^2*C(27*x^6)^(1/3) - x, where C(x) = 1 + x*C(x)^2  = (1-sqrt(1-4*x))/(2*x) is the Catalan function (A000108).

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)).

A370765 a(n) = 9^n * [x^n] Product_{k>=1} ((1 + 2^(k+1)x^k) (1 + 2^(k-1)*x^k))^(1/3).

Original entry on oeis.org

1, 15, 153, 11295, 31968, 5289300, 41957514, 3216919050, -21009764691, 2153132775315, -16978376482767, 1659596014366335, -35929151338082922, 1473739361689662990, -38968782475183427016, 1541715187631618436300, -46858796372722560413526, 1615119529247884664988030

Offset: 0

Vaclav Kotesovec, Mar 01 2024

sign

Cf. A032302, A304961, A370016, A370761, A370764.

nmax = 25; CoefficientList[Series[Product[(1+2^(k+1)*x^k)*(1+2^(k-1)*x^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x] * 9^Range[0, nmax]
nmax = 25; CoefficientList[Series[Product[(1+2^(k+1)*(9*x)^k)*(1+2^(k-1)*(9*x)^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x]
nmax = 25; CoefficientList[Series[(2*QPochhammer[-2, 2*x]*QPochhammer[-1/2, 2*x]/9)^(1/3), {x, 0, nmax}], x] * 9^Range[0, nmax]
nmax = 25; CoefficientList[Series[(2*QPochhammer[-2, x]*QPochhammer[-1/2, x]/9)^(1/3), {x, 0, nmax}], x] * 18^Range[0, nmax]

G.f.: Product_{k>=1} ((1 + 2^(k+1)*(9*x)^k) * (1 + 2^(k-1)*(9*x)^k))^(1/3).

a(n) ~ (-1)^(n+1) * c * 36^n / n^(4/3), where c = 0.244280405759762854740979712556383125782589356973734984...

cp's OEIS Frontend

A370145 Expansion of ( (1 + x)*(1 - 2*x)*(1 + 4*x) )^(1/3).

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A370019 Expansion of [ Sum_{n>=0} (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2) ]^(1/3).

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A370336 Expansion of [ Sum_{n>=0} 5^n * (2*4^n + 1)/3 * x^(n*(n+1)/2) ]^(1/3).

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A370015 Expansion of A(x) = Sum_{n>=0} 2^(n*(n-1)/2) * (1 + 2^(2*n+1))/3 * x^(n*(n+1)/2).

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A370148 Expansion of A(x) = [ Sum_{n>=0} (-7)^(n*(n-1)/2) * (1 + (-7)^(2*n+1))/(-6) * x^(n*(n+1)/2) ]^(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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A370146 Expansion of x / Series_Reversion( x/(1 + 3*x - 6*x^2 - 8*x^3)^(1/3) ).

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

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A370145 Expansion of ( (1 + x)(1 - 2x)(1 + 4x) )^(1/3).

A370019 Expansion of [ Sum_{n>=0} (-4)^n * (24^n + 1)/3 x^(n*(n+1)/2) ]^(1/3).

A370336 Expansion of [ Sum_{n>=0} 5^n * (24^n + 1)/3 x^(n*(n+1)/2) ]^(1/3).

A370015 Expansion of A(x) = Sum_{n>=0} 2^(n(n-1)/2) (1 + 2^(2n+1))/3 x^(n*(n+1)/2).

A370148 Expansion of A(x) = [ Sum_{n>=0} (-7)^(n(n-1)/2) (1 + (-7)^(2n+1))/(-6) x^(n*(n+1)/2) ]^(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).

A370146 Expansion of x / Series_Reversion( x/(1 + 3x - 6x^2 - 8*x^3)^(1/3) ).

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

A370765 a(n) = 9^n * [x^n] Product_{k>=1} ((1 + 2^(k+1)x^k) (1 + 2^(k-1)*x^k))^(1/3).