A370019 -id:A370019 - 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

A370044 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, 32, 240, 2048, 17920, 163904, 1526784, 14473216, 138743808, 1342326528, 13078851584, 128177979392, 1262257356800, 12481163427840, 123845494105088, 1232601926811648, 12300407336042496, 123037059803447296, 1233275751577944064, 12385053557486911488, 124585853452251328512

Offset: 0

Paul D. Hanna, Feb 24 2024

nonn

			G.f.: 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 + 1342326528*x^10 + ...
RELATED SERIES.
The cube of 1/A(x) equals the g.f. A370018 which starts as
1/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) + ...
and 1/A(x) equals the g.f. of A370019, which begins
1/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 + ... + A370019(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..1035

Cf. A370019, A370018, A370045.

{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, 25, print1(a(n), ", "))

a(n) ~ c * d^n / n^(2/3), where d = 10.3933629985595735031515117628403087010816839988881759248638104... and c = 0.42093748110527419326289922348630166534660617909266766696... - Vaclav Kotesovec, Feb 24 2024

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

A370018 Expansion of Sum_{n>=0} (-4)^n * (24^n + 1)/3 x^(n*(n+1)/2).

Original entry on oeis.org

1, -12, 0, 176, 0, 0, -2752, 0, 0, 0, 43776, 0, 0, 0, 0, -699392, 0, 0, 0, 0, 0, 11186176, 0, 0, 0, 0, 0, 0, -178962432, 0, 0, 0, 0, 0, 0, 0, 2863333376, 0, 0, 0, 0, 0, 0, 0, 0, -45813071872, 0, 0, 0, 0, 0, 0, 0, 0, 0, 733008101376, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -11728125427712

Offset: 0

Paul D. Hanna, Feb 23 2024

sign

Equals the self-convolution cube of A370019.

			G.f.: A(x) = 1 - 12*x + 176*x^3 - 2752*x^6 + 43776*x^10 - 699392*x^15 + 11186176*x^21 - 178962432*x^28 + 2863333376*x^36 + ... + (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2) + ...
RELATED SERIES.
The cube root of g.f. A(x) is an integer series starting as
A(x)^(1/3) = 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 + ... + A370019(n)*x^n + ...
Also,
A(x)^(1/6) = 1 - 2*x - 10*x^2 - 44*x^3 - 330*x^4 - 2508*x^5 - 21476*x^6 - 185720*x^7 - 1658778*x^8 - 15042060*x^9 - 138464620*x^10 + ...
The expansion of 1/A(x) begins
1/A(x) = 1 + 12*x + 144*x^2 + 1552*x^3 + 16512*x^4 + 172800*x^5 + 1803200*x^6 + 18765312*x^7 + 195167232*x^8 + 2028914688*x^9 + ... + A370045(n)*x^n + ...
Further,
1/A(x)^(1/3) = 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 + ...

Cf. A370019, A370045, A370044, A370015, A370335.

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

A370045 Expansion of 1 / Sum_{n>=0} (-4)^n * (24^n + 1)/3 x^(n*(n+1)/2).

Original entry on oeis.org

1, 12, 144, 1552, 16512, 172800, 1803200, 18765312, 195167232, 2028914688, 21089678592, 219201730560, 2278287884288, 23679245377536, 246107817345024, 2557891149933568, 26585106479751168, 276308723697205248, 2871777147680423936, 29847423508786839552, 310215112347152351232

Offset: 0

Paul D. Hanna, Feb 24 2024

nonn

			G.f.: A(x) = 1 + 12*x + 144*x^2 + 1552*x^3 + 16512*x^4 + 172800*x^5 + 1803200*x^6 + 18765312*x^7 + 195167232*x^8 + 2028914688*x^9 + 21089678592*x^10 + ...
RELATED SERIES.
The expansion of 1/A(x) is the following series (A370018)
1/A(x) = 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 cube root of A(x) begins
A(x)^(1/3) = 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 + ...
Also, the sixth root of A(x) is an integer series starting as
A(x)^(1/6) = 1 + 2*x + 14*x^2 + 92*x^3 + 742*x^4 + 6188*x^5 + 54956*x^6 + 498584*x^7 + 4625478*x^8 + 43493324*x^9 + 413627172*x^10 + ...

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

Cf. A370018 (1/A(x)), A370044 (A(x)^(1/3)), A370019 (A(x)^(-1/3)).

{a(n) = my(A);A = 1 / sum(m=0, sqrtint(2*n+1), (-4)^m * (1 + 2*4^m)/3 * x^(m*(m+1)/2) +x*O(x^n));polcoeff(H=A, n)}
for(n=0, 25, print1(a(n), ", "))

From Vaclav Kotesovec, Feb 25 2024: (Start)
a(n) ~ c * d^n, where
d = 10.39336299855957350315151176284030870108168399888817592486381041027988779...
c = 1.433973222898078483437999597179822040398973315396494951383570608840342399...
d = 1/r, where r = 0.09621524814812982023560791941974657613430770687333255066... is the smallest positive root of the equation Sum_{k>=0} (-4)^k * (2*4^k + 1) * r^(k*(k+1)/2) = 0. (End)

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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A370044 Expansion of [ Sum_{n>=0} (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2) ]^(-1/3).

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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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A370018 Expansion of Sum_{n>=0} (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2).

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A370045 Expansion of 1 / Sum_{n>=0} (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2).

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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).

A370044 Expansion of [ Sum_{n>=0} (-4)^n * (24^n + 1)/3 x^(n*(n+1)/2) ]^(-1/3).

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).

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

A370018 Expansion of Sum_{n>=0} (-4)^n * (24^n + 1)/3 x^(n*(n+1)/2).

A370045 Expansion of 1 / Sum_{n>=0} (-4)^n * (24^n + 1)/3 x^(n*(n+1)/2).