A370716 -id:A370716 - 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

A370715 a(n) = 3^(2n) [x^n] Product_{k>=1} 1/(1 - 2*x^k)^(1/3).

Original entry on oeis.org

1, 6, 126, 1818, 32130, 452142, 8006526, 117619290, 1999520154, 31550881374, 527781570174, 8556328428786, 145177242834330, 2404855490356782, 40907085509085750, 691705559193384114, 11840743106503713594, 202344257179543757526, 3487245860820904368822, 60077736592697832105330

Offset: 0

Vaclav Kotesovec, Feb 27 2024

nonn

Cf. A070933, A370716.

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

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

a(n) ~ c * 18^n / n^(2/3), where c = 1 / (Gamma(1/3) * QPochhammer(1/2)^(1/3)) = 0.564734286036917647642848904946237...

A370736 a(n) = 4^n * [x^n] Product_{k>=1} (1 + 2*x^k)^(1/4).

Original entry on oeis.org

1, 2, 2, 76, -106, 1788, -1516, 57176, -276634, 2270444, -10094212, 97699752, -664173444, 4819718488, -33236872088, 259931360688, -1894783205754, 13983087008588, -103270227527444, 779496572387208, -5855545477963244, 44016069418771976, -331519650617078376, 2514477954420678352

Offset: 0

Vaclav Kotesovec, Feb 28 2024

sign

Cf. A032302 (m=1), A370709 (m=2), A370716 (m=3), A370737 (m=5).

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

G.f.: Product_{k>=1} (1 + 2*(4*x)^k)^(1/4).

a(n) ~ (-1)^(n+1) * QPochhammer(-1/2)^(1/4) * 8^n / (4 * Gamma(3/4) * n^(5/4)).

A370739 a(n) = 5^(2n) [x^n] Product_{k>=1} (1 + 3*x^k)^(1/5).

Original entry on oeis.org

1, 15, -75, 35250, -1138125, 72645000, -3307996875, 244578890625, -15502648125000, 985908809765625, -63515254624218750, 4314500023927734375, -291905297026816406250, 19789483493484814453125, -1355414138248614990234375, 93666904586649390380859375, -6498800175020013123779296875

Offset: 0

Vaclav Kotesovec, Feb 28 2024

sign

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

Cf. A032308 (d=3,m=1), A370711 (d=3,m=2), A370712 (d=3,m=3), A370738 (d=3,m=4).
Cf. A032302 (d=2,m=1), A370709 (d=2,m=2), A370716 (d=2,m=3), A370736 (d=2,m=4), A370737 (d=2,m=5).
Cf. A000009 (d=1,m=1), A298994 (d=1,m=2), A303074 (d=1,m=3)

nmax = 20; CoefficientList[Series[Product[1+3*x^k, {k, 1, nmax}]^(1/5), {x, 0, nmax}], x] * 25^Range[0, nmax]
nmax = 20; CoefficientList[Series[Product[1+3*(25*x)^k, {k, 1, nmax}]^(1/5), {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + 3*(25*x)^k)^(1/5).

a(n) ~ (-1)^(n+1) * QPochhammer(-1/3)^(1/5) * 75^n / (5 * Gamma(4/5) * n^(6/5)).

A370737 a(n) = 5^(2n) [x^n] Product_{k>=1} (1 + 2*x^k)^(1/5).

Original entry on oeis.org

1, 10, 50, 14750, -166250, 14011250, -133418750, 18136968750, -620089531250, 29520532031250, -917207280468750, 51260806902343750, -2257145499863281250, 101035630688769531250, -4434459153208496093750, 214279556679692871093750, -9859289197933918457031250, 454976266920750451660156250

Offset: 0

Vaclav Kotesovec, Feb 28 2024

sign

Cf. A032302 (m=1), A370709 (m=2), A370716 (m=3), A370736 (m=4).

nmax = 20; CoefficientList[Series[Product[1+2*x^k, {k, 1, nmax}]^(1/5), {x, 0, nmax}], x] * 25^Range[0, nmax]
nmax = 20; CoefficientList[Series[Product[1+2*(25*x)^k, {k, 1, nmax}]^(1/5), {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + 2*(25*x)^k)^(1/5).

a(n) ~ (-1)^(n+1) * QPochhammer(-1/2)^(1/5) * 50^n / (5 * Gamma(4/5) * n^(6/5)).

A370750 a(n) = 9^n * [x^n] Product_{k>=1} ((1 + 2x^k)/(1 - 2x^k))^(1/3).

Original entry on oeis.org

1, 12, 180, 3852, 50436, 947052, 14087844, 245858652, 3531115620, 64019229660, 950199749748, 16959724619004, 256888616329044, 4642974930688812, 71716402072904724, 1308491345357401068, 20501966472318764388, 376230182366985289164, 5987314157007778195716, 110286515004790197907836

Offset: 0

Vaclav Kotesovec, Feb 29 2024

nonn

Cf. A261584, A303346, A370749.
Cf. A032302, A070933.
Cf. A370716, A370715.

nmax = 20; CoefficientList[Series[Product[(1 + 2*x^k)/(1 - 2*x^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x] * 9^Range[0, nmax]
nmax = 20; CoefficientList[Series[Product[(1 + 2*(9*x)^k)/(1 - 2*(9*x)^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x]

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

a(n) ~ QPochhammer(-1, 1/2)^(1/3) * 18^n / (Gamma(1/3) * QPochhammer(1/2)^(1/3) * n^(2/3)).

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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A370715 a(n) = 3^(2*n) * [x^n] Product_{k>=1} 1/(1 - 2*x^k)^(1/3).

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A370736 a(n) = 4^n * [x^n] Product_{k>=1} (1 + 2*x^k)^(1/4).

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A370739 a(n) = 5^(2*n) * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/5).

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A370737 a(n) = 5^(2*n) * [x^n] Product_{k>=1} (1 + 2*x^k)^(1/5).

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A370750 a(n) = 9^n * [x^n] Product_{k>=1} ((1 + 2*x^k)/(1 - 2*x^k))^(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).

A370715 a(n) = 3^(2n) [x^n] Product_{k>=1} 1/(1 - 2*x^k)^(1/3).

A370739 a(n) = 5^(2n) [x^n] Product_{k>=1} (1 + 3*x^k)^(1/5).

A370737 a(n) = 5^(2n) [x^n] Product_{k>=1} (1 + 2*x^k)^(1/5).

A370750 a(n) = 9^n * [x^n] Product_{k>=1} ((1 + 2x^k)/(1 - 2x^k))^(1/3).