A298994 -id:A298994 - cp's OEIS frontend

A303074 Expansion of Product_{n>=1} (1 + (9*x)^n)^(1/3).

1, 3, 18, 369, 1674, 31428, 266733, 3012714, 19924299, 319970007, 2688208641, 27248985549, 248061612240, 2597556114648, 25367004717831, 289880288735373, 2289952155529719, 23895509092285545, 252143223166599723, 2308267172943599733, 22389894059315522040

Offset: 0

Vaclav Kotesovec, Apr 18 2018

nonn

In general, for h>=1, if g.f. = Product_{k>=1} (1 + (h^2*x)^k)^(1/h), then a(n) ~ h^(2*n) * exp(Pi*sqrt(n/(3*h))) / (2^((3*h + 1)/(2*h)) * 3^(1/4) * h^(1/4) * n^(3/4)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A271236, A298994, A303342, A370739.

CoefficientList[Series[(QPochhammer[-1, 9*x]/2)^(1/3), {x, 0, 20}], x]

a(n) ~ 3^(2*n - 1/2) * exp(sqrt(n)*Pi/3) / (2^(5/3) * n^(3/4)).

A271235 G.f. equals the square root of P(4*x), where P(x) is the g.f. of the partition numbers (A000041).

Original entry on oeis.org

1, 2, 14, 68, 406, 1820, 10892, 48008, 266214, 1248044, 6454116, 29642424, 156638076, 707729176, 3551518936, 16671232784, 81685862790, 375557689292, 1843995831412, 8437648295384, 40779718859796, 188104838512840, 891508943457064, 4091507664092016, 19457793452994012, 88760334081132280, 415942096027738728, 1905990594266105648, 8875964207106121784, 40416438507461834160

Offset: 0

Paul D. Hanna, Apr 02 2016

nonn

More formulas and information can be derived from entry A000041.

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/2, g(n) = 4^n. - Seiichi Manyama, Apr 20 2018

			G.f.: A(x) = 1 + 2*x + 14*x^2 + 68*x^3 + 406*x^4 + 1820*x^5 + 10892*x^6 + 48008*x^7 + 266214*x^8 + 1248044*x^9 + 6454116*x^10 +...
where A(x)^2 = P(4*x).
RELATED SERIES.
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + 42*x^10 + 56*x^11 + 77*x^12 + 101*x^13 + 135*x^14 +...+ A000041(n)*x^n +...
1/A(x)^6 = 1 - 12*x + 320*x^3 - 28672*x^6 + 9437184*x^10 - 11811160064*x^15  + 57174604644352*x^21 +...+ (-1)^n*(2*n+1)*(4*x)^(n*(n+1)/2) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A298411, A298993, A298994, A370735.

Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(-1/b): A000041 (b=1), this sequence (b=2), A271236 (b=3), A303135 (b=4), A303136 (b=5).

nmax = 30; CoefficientList[Series[Product[1/Sqrt[1 - (4*x)^k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 02 2016 *)

{a(n) = polcoeff( prod(k=1,n, 1/sqrt(1 - (4*x)^k +x*O(x^n))),n)}
for(n=0,30,print1(a(n),", "))

{a(n) = if( n<0, 0, polcoeff( sum( k=1, sqrtint(n+1), (4*x)^(k^2) / prod(j=1, k, 1 - (4*x)^j, 1 + x*O(x^n))^2, 1)^(1/2), n))};
for(n=0,30,print1(a(n),", "))

N=99; x='x+O('x^N); Vec(prod(k=1, N, 1/(1-(4*x)^k)^(1/2))) \\ Altug Alkan, Apr 20 2018

G.f.: Product_{n>=1} 1/sqrt(1 - (4*x)^n).
Sum_{k=0..n} a(k) * a(n-k)  =  4^n * A000041(n), for n>=0, where A000041(n) equals the number of partitions of n.
a(n) ~ 4^(n-1) * exp(sqrt(n/3)*Pi) / (3^(3/8) * n^(7/8)). - Vaclav Kotesovec, Apr 02 2016

A271236 G.f.: Product_{k>=1} 1/(1 - (9*x)^k)^(1/3).

Original entry on oeis.org

1, 3, 45, 450, 5805, 52326, 705591, 6190425, 77219325, 751178610, 8522919063, 80502824835, 975122402985, 8949951461925, 100088881882830, 1003346683458480, 10828622925516312, 104307212166072165, 1152197107898173875, 11048535008792967825, 119509353627934830327

Offset: 0

Vaclav Kotesovec, Apr 02 2016

nonn

In general, for h>=1, if g.f. = Product_{k>=1} 1/(1-(h^2*x)^k)^(1/h), then a(n) ~ h^(2*n) * exp(Pi*sqrt(2*n/(3*h))) / (2^(5*h+3) * 3^(h+1) * h^(h+1) * n^(3*h+1))^(1/(4*h)).

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/3, g(n) = 9^n. - Seiichi Manyama, Apr 20 2018

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A298994, A303074, A303130, A303152, A303342, A370735.

Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(-1/b): A000041 (b=1), A271235 (b=2), this sequence (b=3), A303135 (b=4), A303136 (b=5).

nmax = 30; CoefficientList[Series[Product[1/(1 - (9*x)^k)^(1/3), {k, 1, nmax}], {x, 0, nmax}], x]

N=99; x='x+O('x^N); Vec(prod(k=1, N, 1/(1-(9*x)^k)^(1/3))) \\ Altug Alkan, Apr 20 2018

a(n) ~ 3^(2*n - 2/3) * exp(sqrt(2*n)*Pi/3) / (2^(3/2) * n^(5/6)).

A298411 Coefficients of q^(-1/24)*eta(4q)^(1/2).

Original entry on oeis.org

1, -2, -10, -20, -90, 132, -836, 6040, 2310, 60180, 180308, 1662568, -2995620, 24401320, 44072120, -102437328, 19390406, 2649221300, -10584460060, 14475802440, -228570333836, -815899620616, 2088529753800, -5590702681520, -100828534100580, -172013432412024

Offset: 0

William J. Keith, Jan 18 2018

The q^(kn) term of any single factor of the product (1-(4q)^k)^(1/2) is (-2)*A000108(n-1). Hence these numbers are related to the Catalan numbers A000108 by a partition-based convolution.
Sequence appears to be positive and negative roughly half the time.
This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/2, g(n) = 4^n. - Seiichi Manyama, Apr 20 2018

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000108, A271235, A298994.

Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(1/b): A010815 (b=1), this sequence (b=2), A303152 (b=3), A303153 (b=4), A303154 (b=5).

Series[Product[(1 - (4 q)^k)^(1/2), {k, 1, 100}], {q, 0, 100}]

q='q+O('q^99); Vec(eta(4*q)^(1/2)) \\ Altug Alkan, Apr 20 2018

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

A298993 Expansion of Product_{n>=1} 1/sqrt(1 + (4*x)^n).

Original entry on oeis.org

1, -2, -2, -36, 54, -476, 556, -6088, 35878, -156844, 444164, -1734648, 11948604, -35313048, 156354328, -864527760, 4733447686, -12692853452, 54065039380, -226098757912, 1278838329812, -5257771138376, 19455009120232, -76455773381360, 453306681446748

Offset: 0

Seiichi Manyama, Jan 31 2018

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000984, A081362, A271235, A298994.

CoefficientList[Series[Sqrt[2/QPochhammer[-1, 4*x]], {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 18 2018 *)

a(n) ~ (-1)^n * exp(Pi*sqrt(n/12)) * 4^(n-1) / (3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 18 2018

A370709 a(n) = 2^n * [x^n] Product_{k>=1} (1 + 2*x^k)^(1/2).

Original entry on oeis.org

1, 2, 2, 20, 6, 108, 148, 776, -186, 5964, -4, 51032, -89700, 512120, -1259416, 6406032, -19733434, 78363148, -268823572, 1047941688, -3800035916, 14327505832, -52766730600, 199492430192, -746479735524, 2811936761016, -10588174502568, 40092283176560, -151796846803592

Offset: 0

Vaclav Kotesovec, Feb 27 2024

sign

Cf. A032302, A370713.

Cf. A075900, A298994, A300581, A304961, A327550.

nmax = 30; CoefficientList[Series[Product[(1 + 2*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x] * 2^Range[0, nmax]
nmax = 30; CoefficientList[Series[Product[(1 + 2*(2*x)^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Sqrt[QPochhammer[-2, x]/3], {x, 0, nmax}], x] * 2^Range[0, nmax]

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

a(n) ~ (-1)^(n+1) * c * 4^n / n^(3/2), where c = QPochhammer(-1/2)^(1/2) / (2*sqrt(Pi)) = 0.31039710860287467176143051675437...

A303124 Expansion of Product_{n>=1} (1 + (16*x)^n)^(1/4).

Original entry on oeis.org

1, 4, 40, 1504, 10336, 387968, 5349632, 111442944, 1100563968, 36711258112, 493805416448, 9186633203712, 134635599806464, 2648342619422720, 43443234834350080, 938422838970810368, 11378951438668791808, 224791017150689574912, 4129154423023897411584

Offset: 0

Seiichi Manyama, Apr 19 2018

nonn

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/4, g(n) = -16^n.

Seiichi Manyama, Table of n, a(n) for n = 0..500

Expansion of Product_{n>=1} (1 + ((b^2)*x)^n)^(1/b): A000009 (b=1), A298994 (b=2), A303074 (b=3), this sequence (b=4), A303125 (b=5).

Cf. A303131, A303153.

CoefficientList[Series[(QPochhammer[-1, 16*x]/2)^(1/4), {x, 0, 20}],
x] (* Vaclav Kotesovec, Apr 19 2018 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+(16*x)^k)^(1/4)))

a(n) ~ 2^(4*n - 17/8) * exp(sqrt(n/3)*Pi/2) / (3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 19 2018

A303125 Expansion of Product_{n>=1} (1 + (25*x)^n)^(1/5).

Original entry on oeis.org

1, 5, 75, 4500, 43125, 2765000, 55871875, 1876671875, 25128437500, 1495793359375, 28953471875000, 871257974609375, 18280647500000000, 596362168603515625, 14502797130615234375, 519397373566650390625, 8604439235863037109375

Offset: 0

Seiichi Manyama, Apr 19 2018

nonn

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/5, g(n) = -25^n.

Seiichi Manyama, Table of n, a(n) for n = 0..500

Expansion of Product_{n>=1} (1 + ((b^2)*x)^n)^(1/b): A000009 (b=1), A298994 (b=2), A303074 (b=3), A303124 (b=4), this sequence (b=5).

Cf. A303132, A303154.

CoefficientList[Series[(QPochhammer[-1, 25*x]/2)^(1/5), {x, 0, 20}],
x] (* Vaclav Kotesovec, Apr 19 2018 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+(25*x)^k)^(1/5)))

a(n) ~ 5^(2*n - 1/4) * exp(Pi*sqrt(n/15)) / (2^(8/5) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 19 2018

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)).

cp's OEIS Frontend

A303074 Expansion of Product_{n>=1} (1 + (9*x)^n)^(1/3).

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A271235 G.f. equals the square root of P(4*x), where P(x) is the g.f. of the partition numbers (A000041).

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A271236 G.f.: Product_{k>=1} 1/(1 - (9*x)^k)^(1/3).

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A298411 Coefficients of q^(-1/24)*eta(4q)^(1/2).

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A298993 Expansion of Product_{n>=1} 1/sqrt(1 + (4*x)^n).

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A370709 a(n) = 2^n * [x^n] Product_{k>=1} (1 + 2*x^k)^(1/2).

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A303124 Expansion of Product_{n>=1} (1 + (16*x)^n)^(1/4).

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A303125 Expansion of Product_{n>=1} (1 + (25*x)^n)^(1/5).

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