A271236 -id:A271236 - 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

A303135 Expansion of Product_{n>=1} (1 - (16*x)^n)^(-1/4).

Original entry on oeis.org

1, 4, 104, 1760, 39520, 590720, 14285056, 205151232, 4596467200, 75375073280, 1504196046848, 23673049726976, 525315968712704, 7912159583600640, 158055039529779200, 2726833423421800448, 51889395654107463680, 840470097284214292480, 16765991910040314839040

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): A000041 (b=1), A271235 (b=2), A271236 (b=3), this sequence (b=4), A303136 (b=5).

Cf. A303131, A303153.

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

a(n) ~ exp(sqrt(n/6)*Pi) * 2^(4*n - 33/16) / (3^(5/16) * n^(13/16)). - Vaclav Kotesovec, Apr 19 2018

A303136 Expansion of Product_{n>=1} (1 - (25*x)^n)^(-1/5).

Original entry on oeis.org

1, 5, 200, 5125, 177500, 3952500, 150715625, 3185187500, 112844843750, 2783033593750, 86330708203125, 2019237027343750, 72195817812500000, 1591910699609375000, 50158322275878906250, 1322261581989501953125, 39183430287559814453125, 946961406814801025390625

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): A000041 (b=1), A271235 (b=2), A271236 (b=3), A303135 (b=4), this sequence (b=5).

Cf. A303132, A303154.

CoefficientList[Series[1/QPochhammer[25*x]^(1/5), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 19 2018 *)
CoefficientList[Series[Product[(1-(25x)^n)^(-1/5),{n,20}],{x,0,20}],x] (* Harvey P. Dale, Nov 04 2021 *)

a(n) ~ exp(Pi*sqrt(2*n/15)) * 5^(2*n - 3/10) / (2^(7/5) * 3^(3/10) * n^(4/5)). - Vaclav Kotesovec, Apr 19 2018

A303152 Expansion of Product_{n>=1} (1 - (9*x)^n)^(1/3).

Original entry on oeis.org

1, -3, -36, -207, -2214, -2754, -138591, 547722, -3730293, 30138075, 133709535, 7735237479, -35284817430, 702841889322, 3056530613769, 9493893988155, 112554319443867, 3822223052352735, -3940051663965051, 250298859930263181, -551418001934739786, 1061747224529191191

Offset: 0

Seiichi Manyama, Apr 19 2018

sign

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

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

Expansion of Product_{n>=1} (1 - ((b^2)*x)^n)^(1/b): A010815 (b=1), A298411 (b=2), this sequence (b=3), A303153 (b=4), A303154 (b=5).

Cf. A271236, A303074.

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-(9*x)^k)^(1/3)))

A303342 Expansion of Product_{k>=1} ((1 + (9x)^k) / (1 - (9x)^k))^(1/3).

Original entry on oeis.org

1, 6, 72, 1008, 10746, 130896, 1569456, 17371584, 192625128, 2260005462, 24725148912, 270748885392, 3027318848208, 32608207056528, 354309508944288, 3902606972751168, 41393526342215994, 443390745816982944, 4783687280410092984, 50532141192366275280

Offset: 0

Vaclav Kotesovec, Apr 22 2018

nonn

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

Cf. A271236, A303074, A303307.

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

a(n) ~ 3^(2*n) * exp(Pi*sqrt(n/3)) / (4 * 3^(1/3) * n^(5/6)).

A370735 a(n) = 5^(2n) [x^n] Product_{k>=1} 1/(1 - 3*x^k)^(1/5).

Original entry on oeis.org

1, 15, 1050, 52125, 3277500, 179801250, 11966690625, 738318187500, 49788716718750, 3314446448437500, 227432073022265625, 15631633385109375000, 1090877899335878906250, 76338563689129101562500, 5384934139819611328125000, 381204340327212964599609375, 27111589537137988341064453125

Offset: 0

Vaclav Kotesovec, Feb 28 2024

nonn

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

Cf. A242587 (d=3,m=1), A370714 (d=3,m=2), A370710 (d=3,m=3), A370734 (d=3,m=4).
Cf. A070933 (d=2,m=1), A370713 (d=2,m=2), A370715 (d=2,m=3), A370732 (d=2,m=4), A370733 (d=2,m=5).
Cf. A000041 (d=1,m=1), A271235 (d=1,m=2), A271236 (d=1,m=3).

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

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

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

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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A303135 Expansion of Product_{n>=1} (1 - (16*x)^n)^(-1/4).

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A303136 Expansion of Product_{n>=1} (1 - (25*x)^n)^(-1/5).

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A303152 Expansion of Product_{n>=1} (1 - (9*x)^n)^(1/3).

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A303342 Expansion of Product_{k>=1} ((1 + (9*x)^k) / (1 - (9*x)^k))^(1/3).

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A370735 a(n) = 5^(2*n) * [x^n] Product_{k>=1} 1/(1 - 3*x^k)^(1/5).

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A303342 Expansion of Product_{k>=1} ((1 + (9x)^k) / (1 - (9x)^k))^(1/3).

A370735 a(n) = 5^(2n) [x^n] Product_{k>=1} 1/(1 - 3*x^k)^(1/5).