A370711 -id:A370711

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page 1

A370714

a(n) = 4^n * [x^n] Product_{k>=1} 1/(1 - 3*x^k)^(1/2).

+10
4

1, 6, 78, 780, 8790, 90708, 1015692, 10964760, 122893926, 1370476932, 15518261220, 176063641512, 2014426860540, 23109736996680, 266397931733208, 3079014279154224, 35695144493030022, 414708043501061988, 4828444403991450612, 56314242827277224712, 657855733949279381652

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..20.

FORMULA

G.f.: Product_{k>=1} 1/(1 - 3*(4*x)^k)^(1/2).

a(n) ~ 12^n / sqrt(Pi*QPochhammer(1/3)*n).

MATHEMATICA

nmax = 25; CoefficientList[Series[Product[1/(1-3*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x] * 4^Range[0, nmax]

nmax = 25; CoefficientList[Series[Product[1/(1-3*(4*x)^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x]

nmax = 25; CoefficientList[Series[Sqrt[-2/QPochhammer[3, x]], {x, 0, nmax}], x] * 4^Range[0, nmax]

CROSSREFS

Cf. A242587, A370711.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Feb 27 2024

STATUS

approved

A370739

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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)).

LINKS

Table of n, a(n) for n=0..16.

FORMULA

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)).

MATHEMATICA

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]

CROSSREFS

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)

KEYWORD

sign

AUTHOR

Vaclav Kotesovec, Feb 28 2024

STATUS

approved

A370738

a(n) = 8^n * [x^n] Product_{k>=1} (1 + 3*x^k)^(1/4).

+10
1

1, 6, -6, 1428, -13146, 280788, -3785820, 93142824, -1851272826, 37533646212, -765409050420, 16617464296728, -357906128318628, 7730398360992840, -168750405673899000, 3719099270015849040, -82288133754592611642, 1828585054153956768612, -40828782977534929747524, 915461326204911371035320

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..19.

FORMULA

G.f.: Product_{k>=1} (1 + 3*(8*x)^k)^(1/4).

a(n) ~ (-1)^(n+1) * QPochhammer(-1/3)^(1/4) * 24^n / (4 * Gamma(3/4) * n^(5/4)).

MATHEMATICA

nmax = 20; CoefficientList[Series[Product[1+3*x^k, {k, 1, nmax}]^(1/4), {x, 0, nmax}], x] * 8^Range[0, nmax]

nmax = 20; CoefficientList[Series[Product[1+3*(8*x)^k, {k, 1, nmax}]^(1/4), {x, 0, nmax}], x]

CROSSREFS

Cf. A032308 (m=1), A370711 (m=2), A370712 (m=3), A370739 (m=5).

KEYWORD

sign

AUTHOR

Vaclav Kotesovec, Feb 28 2024

STATUS

approved

A370751

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

+10
1

1, 6, 30, 204, 966, 5748, 29388, 169944, 886278, 5169732, 27794820, 162920616, 894445212, 5274022920, 29398573272, 174041671344, 980746798278, 5821525480164, 33071756442708, 196663513473672, 1124154722216244, 6693497121210648, 38448301937075112, 229149691659210192

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..23.

FORMULA

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

a(n) ~ c * 6^n / n^(1/2), where c = (QPochhammer(-1,1/3) / (Pi * QPochhammer(1/3)))^(1/2) = 1.333660169175690343841707335109800906849893636...

MATHEMATICA

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

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

CROSSREFS