A304443 -id:A304443 - cp's OEIS frontend

A304444 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^(2*n).

1, 2, 14, 98, 726, 5512, 42614, 333608, 2636326, 20985272, 168012824, 1351507830, 10914317934, 88432329546, 718545161208, 5852747363518, 47774241056710, 390702055798978, 3200542803221192, 26257321971526646, 215705170816632376, 1774181109262878848

Offset: 0

Vaclav Kotesovec, May 12 2018

nonn

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

Cf. A000712, A008485, A161870, A171803, A304443.

nmax = 25; Table[SeriesCoefficient[Product[1/(1-x^k)^(2*n), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
nmax = 25; Table[SeriesCoefficient[1/QPochhammer[x]^(2*n), {x, 0, n}], {n, 0, nmax}]
(* Calculation of constants {d,c}: *) eq = FindRoot[{1/QPochhammer[r*s]^2 == s, 1/s + 2*r*Sqrt[s]*Derivative[0, 1][QPochhammer][r*s, r*s] == (2*(Log[1 - r*s] + QPolyGamma[0, 1, r*s]))/(s* Log[r*s])}, {r, 1/8}, {s, 2}, WorkingPrecision -> 1000]; {N[1/r /. eq, 120], val = Sqrt[((1 - r*s)*Log[r*s]^2)/(Pi*(16*r*s*ArcTanh[1 - 2*r*s] - (-1 + r*s)*(Log[r*s] - 2*Log[1 - r*s])*(3*Log[r*s] - 2*Log[1 - r*s]) - 8*Log[1 - r*s] - 8*(-1 + r*s)*(-1 + 2*ArcTanh[1 - 2*r*s])* QPolyGamma[0, 1, r*s] + (4 - 4*r*s)* QPolyGamma[0, 1, r*s]^2 + 4*(-1 + r*s)*(QPolyGamma[1, 1, r*s] + r*s*Log[r*s] * (r*s^(3/2)*Log[r*s]* Derivative[0, 2][QPochhammer][r*s, r*s] - 2*Derivative[0, 0, 1][QPolyGamma][0, 1, r*s]))))] /. eq; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3]} (* Vaclav Kotesovec, Oct 03 2023 *)

a(n) ~ c * d^n / sqrt(n), where d = 8.42516721063251541777601555584151410936132980324698494327338254953123205... and c = 0.29923152009652750283923119244187982714171590056794904644563876...

A327214 Self-convolution of A270913.

Original entry on oeis.org

1, 2, 7, 32, 137, 592, 2597, 11442, 50567, 224112, 995392, 4428372, 19727877, 87983202, 392755207, 1754625632, 7844003907, 35086658052, 157023432677, 703037135122, 3148915010832, 14108913792342, 63235380631747, 283495965998772, 1271282293531077, 5702105357347602

Offset: 0

Vaclav Kotesovec, Aug 26 2019

nonn

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

Cf. A022567, A270913, A304443, A309986, A327215.

A270913[n_]:=SeriesCoefficient[Product[(1+x^k)^n, {k, 1, n}], {x, 0, n}];
Table[Sum[A270913[k]*A270913[n-k], {k, 0, n}], {n, 0, 25}]

a(n) ~ c^2 * Pi * d^n, where d = A270914 = 4.5024767476173544877385939327... and c = A327280 = 0.260542233142438469433860832160... (see A270913).

A304447 Coefficient of x^n in Product_{k>=1} ((1+x^k)/(1-x^k))^(2*n).

Original entry on oeis.org

1, 4, 40, 448, 5264, 63624, 783328, 9770240, 123040288, 1561033348, 19922193200, 255472920256, 3289122824000, 42488488508808, 550435283089088, 7148519205631488, 93038785849116736, 1213215382135324680, 15846906866928513736, 207302985358274247104

Offset: 0

Vaclav Kotesovec, May 12 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..880

Cf. A261386, A270919, A304443, A304444, A304448.

nmax = 20; Table[SeriesCoefficient[Product[((1+x^k)/(1-x^k))^(2*n), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
nmax = 20; Table[SeriesCoefficient[(QPochhammer[-1, x]/2/QPochhammer[x])^(2*n), {x, 0, n}], {n, 0, nmax}]
(* Calculation of constants {d,c}: *) eq = FindRoot[{QPochhammer[-1, r*s] == 2*Sqrt[s]*QPochhammer[r*s], (QPochhammer[ r*s]*(Log[r*s] - 2*Log[1 - r*s] - 2*QPolyGamma[0, 1, r*s])) / Log[r*s] - r*Sqrt[s]*Derivative[0, 1][QPochhammer][-1, r*s] + 2*r*s*Derivative[0, 1][QPochhammer][r*s, r*s] == 0}, {r, 1/12}, {s, 2}, WorkingPrecision -> 1000]; {N[1/r /. eq, 120], val = Sqrt[((1 - r*s)*Log[r*s]^2*QPochhammer[r*s]) / (Pi*(2*r*s*(-1 + r*s) * Log[r*s]*(2*(Log[r*s] - 2*Log[1 - r*s] - 2*QPolyGamma[0, 1, r*s]) * Derivative[0, 1][QPochhammer][r*s, r*s] + r*Sqrt[s]*Log[r*s] * (-Derivative[0, 2][QPochhammer][-1, r*s] + 2*Sqrt[s]*Derivative[0, 2][QPochhammer][r*s, r*s])) + QPochhammer[ r*s]*(16*r*s*ArcTanh[1 - 2*r*s] + (1 - r*s)*Log[r*s]^2 - 8*Log[1 - r*s] + 4*(-1 + r*s)*Log[1 - r*s]^2 + 8*(-1 + r*s)*(1 + Log[1 - r*s])* QPolyGamma[0, 1, r*s] + 4*(-1 + r*s)*QPolyGamma[0, 1, r*s]^2 + 4*(-1 + r*s)*(QPolyGamma[1, 1, r*s] - 2*r*s*Log[r*s]*Derivative[0, 0, 1][QPolyGamma][0, 1, r*s]))))] /. eq; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3]} (* Vaclav Kotesovec, Oct 03 2023 *)

a(n) ~ c * d^n / sqrt(n), where d = 13.43567525239504624062504283058713960962824709850658926621911428148173077464... and c = 0.3323527904383991069791889982282236666403568774227549868882810268779...

cp's OEIS Frontend

A304444 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^(2*n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A327214 Self-convolution of A270913.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A304447 Coefficient of x^n in Product_{k>=1} ((1+x^k)/(1-x^k))^(2*n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula