A161870 -id:A161870 - 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...

A276551 Convolution square of A073592.

Original entry on oeis.org

1, -2, -3, 2, 6, 12, 1, -10, -32, -46, -24, 18, 96, 168, 213, 124, -61, -386, -734, -992, -957, -386, 685, 2288, 3939, 5158, 5012, 2930, -1853, -8888, -17283, -24782, -28312, -24422, -9825, 16674, 54197, 96584, 134729, 153718, 138624, 73438, -49526, -228614

Offset: 0

Seiichi Manyama, Apr 10 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Column k=2 of A276554.

Product_{k>0} (1-x^k)^(k*m): A161870 (m=-2), A073592 (m=1), this sequence (m=2), A276552 (m=3).

G.f.: Product_{k>0} (1-x^k)^(k*2).

G.f.: exp(-2*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 30 2018

A261384 Expansion of Product_{k>=1} (1+x^k)^(2k-1) / (1-x^k)^(2k).

Original entry on oeis.org

1, 3, 12, 39, 117, 331, 893, 2307, 5766, 13986, 33046, 76302, 172567, 383013, 835731, 1795236, 3801105, 7941439, 16386777, 33423342, 67435311, 134675784, 266385932, 522135379, 1014643823, 1955656848, 3740191268, 7100290646, 13383997996, 25058666367

Offset: 0

Vaclav Kotesovec, Aug 17 2015

nonn

Convolution of A161870 and A255835.

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A006950, A015128, A156616, A161870, A255835, A261386.

nmax = 40; CoefficientList[Series[Product[(1+x^k)^(2*k-1)/(1-x^k)^(2*k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ (7*Zeta(3))^(2/9) * exp(1/6 - Pi^4/(6048*Zeta(3)) - Pi^2 * n^(1/3) / (12*(7*Zeta(3))^(1/3)) + 3/2*(7*Zeta(3))^(1/3) * n^(2/3)) / (A^2 * 2^(1/6) * sqrt(3*Pi) * n^(13/18)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A321947 Column k=2 of triangle A257673.

Original entry on oeis.org

1, 6, 21, 62, 162, 396, 917, 2036, 4380, 9152, 18694, 37380, 73444, 141918, 270370, 508178, 943876, 1733468, 3151396, 5674152, 10126435, 17921016, 31468623, 54848750, 94935565, 163232096, 278903915, 473693432, 799949111, 1343550666, 2244807927, 3731885232

Offset: 2

Alois P. Heinz, Nov 22 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..10000

Column k=2 of A257673.

Cf. A000219, A161870.

b:= proc(n, k) option remember; `if`(n=0, 1, k*add(
      b(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
    end:
a:= n-> (k-> add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(2):
seq(a(n), n=2..35);

A321947[n_] := Module[{nn = n}, SeriesCoefficient[Product[1/(1 - x^i)^(2 i), {i, 1, nn}], {x, 0, nn}] - 2*SeriesCoefficient[Product[(1 - x^k)^-k, {k, nn}], {x, 0, nn}]]; Table[A321947[n], {n, 2, 33}] (* Robert P. P. McKone, Jan 30 2021 *)
b[n_, k_] := b[n, k] = If[n == 0, 1, k*Sum[
     b[n - j, k]*DivisorSigma[2, j], {j, 1, n}]/n];
a[n_] := With[{k = 2}, Sum[b[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]];
Table[a[n], {n, 2, 35}] (* Jean-François Alcover, Aug 23 2021, after Alois P. Heinz *)

G.f.: (-1 + Product_{k>=1} 1 / (1 - x^k)^k)^2. - Ilya Gutkovskiy, Jan 30 2021

a(n) = A161870(n) - 2*A000219(n). - Vaclav Kotesovec, Jan 30 2021

A383353 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where n 2-colored objects are distributed into k containers of two kinds. Containers may be left empty.

Original entry on oeis.org

1, 2, 0, 3, 4, 0, 4, 8, 6, 0, 5, 12, 22, 8, 0, 6, 16, 38, 40, 10, 0, 7, 20, 54, 92, 73, 12, 0, 8, 24, 70, 144, 196, 112, 14, 0, 9, 28, 86, 196, 354, 376, 172, 16, 0, 10, 32, 102, 248, 512, 760, 678, 240, 18, 0, 11, 36, 118, 300, 670, 1200, 1554, 1136, 335, 20, 0

Offset: 0

Peter Dolland, Apr 24 2025

			Array starts:
 0 : [1,  2,   3,    4,     5,     6,     7,      8,      9,     10,     11, ...]
 1 : [0,  4,   8,   12,    16,    20,    24,     28,     32,     36,     40, ...]
 2 : [0,  6,  22,   38,    54,    70,    86,    102,    118,    134,    150, ...]
 3 : [0,  8,  40,   92,   144,   196,   248,    300,    352,    404,    456, ...]
 4 : [0, 10,  73,  196,   354,   512,   670,    828,    986,   1144,   1302, ...]
 5 : [0, 12, 112,  376,   760,  1200,  1640,   2080,   2520,   2960,   3400, ...]
 6 : [0, 14, 172,  678,  1554,  2640,  3810,   4980,   6150,   7320,   8490, ...]
 7 : [0, 16, 240, 1136,  2936,  5436,  8272,  11228,  14184,  17140,  20096, ...]
 8 : [0, 18, 335, 1826,  5315, 10674, 17216,  24262,  31473,  38684,  45895, ...]
 9 : [0, 20, 440, 2812,  9136, 19984, 34192,  50248,  67024,  84020, 101016, ...]
10 : [0, 22, 578, 4186, 15188, 36024, 65512, 100488, 138188, 176878, 215854, ...]
...

Alois P. Heinz, Rows n = 0..200, flattened

Antidiagonal sums give A161870.
Cf. A383351, A383352.
Cf. A382345 (1-color), A381891 (1-kind), A026820 (1-color, 1-kind).
Cf. A278710.

b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, (n+1)*x^n,
      add(b(n-i*j, min(n-i*j, i-1))*binomial(i+j, j)*x^j, j=0..n/i)))
    end:
g:= proc(n, k) option remember;
      `if`(k<0, 0, g(n, k-1)+coeff(b(n$2), x, k))
    end:
A:= (n, k)-> add(add(g(j, h)*g(n-j, k-h), h=0..k), j=0..n):
seq(seq(A(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, May 05 2025

from sympy import binomial
from sympy.utilities.iterables import partitions
def calc_w( k , m):
    s = 0
    for p in partitions( m, m=k+1):
        fact = 1
        j = k + 1
        for x in p :
            fact *= binomial( j, p[x]) * (x + 1) ** p[x]
            j -= p[x]
        s += fact
    return s
def a_row( n, length=11):
    if n == 0 : return [ k + 1 for k in range( length) ]
    t = list( [0] * length)
    for p in partitions( n):
        fact = 1
        s = 0
        for k in p :
            s += p[k]
            fact *= calc_w( k, p[k])
        if s > 0 :
            t[s - 1] += fact
    t = [0] + t
    for i in range( 1, length):
        t[i+1] += t[i] * 2 - t[i - 1]
    return t

A(0,k) = k + 1.
A(1,k) = 4*k.
A(2,k+1) = 6 + 16 * k.
A(n,1) = 2 + 2 * n.
A(n,n+k) = A(n,n) + k * A383352(n,n).
A(n,k) = Sum_{i=0..k} (k + 1 - i) * A383351(n,i) for 0 <= k <= n.
Sum_{k=0..n} (-1)^k*T(n-k,k) = A278710(n). - Alois P. Heinz, May 05 2025

A295828 Expansion of Product_{k>=1} 1/(1 - x^k)^(2k(2*k-1)).

Original entry on oeis.org

1, 2, 15, 58, 235, 862, 3122, 10664, 35639, 115164, 363806, 1122050, 3393316, 10068006, 29374056, 84347944, 238713339, 666419456, 1836986443, 5003473866, 13476019215, 35912177618, 94746481999, 247597696802, 641205816641, 1646268490598, 4192059724668, 10590937903412, 26556243826240

Offset: 0

Ilya Gutkovskiy, Nov 28 2017

nonn

Euler transform of A002939.

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms

Cf. A002939, A161870, A253289, A258347, A258348, A278767.

nmax = 28; CoefficientList[Series[Product[1/(1 - x^k)^(2 k (2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[2 d^2 (2 d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 28}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A002939(k).

a(n) ~ exp(2^(5/2) * Pi * n^(3/4) / (3^(5/4) * 5^(1/4)) - Zeta(3) * sqrt(15*n) / Pi^2 - 15^(5/4) * Zeta(3)^2 * n^(1/4) / (2^(3/2) * Pi^5) - Zeta(3) / Pi^2 - 75*Zeta(3)^3 / (2*Pi^8) - 1/6) * A^2 / (2^(4/3) * 15^(1/12) * Pi^(1/6) * n^(7/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 28 2017

A360489 Convolution of A000219 and A001477.

Original entry on oeis.org

0, 1, 3, 8, 19, 43, 91, 187, 369, 711, 1335, 2459, 4442, 7904, 13851, 23965, 40958, 69248, 115872, 192097, 315652, 514485, 832112, 1336214, 2131099, 3377178, 5319290, 8330147, 12973662, 20100411, 30986772, 47542096, 72609729, 110410791, 167186826, 252138816, 378781852

Offset: 0

Vaclav Kotesovec, Feb 09 2023

nonn

In general, for 0 < p < 1, delta > 1, beta > -1, the convolution of (delta^(n^p) * n^alfa) and n^beta is asymptotic to delta^(n^p) * n^(alfa + (1-p)*(beta+1)) * Gamma(beta+1) / (p^(beta+1) * log(delta)^(beta+1)).

For p = 1 is the convolution of (delta^(n^p) * n^alfa) and n^beta asymptotic to delta^n * n^alfa * polylog(-beta, 1/delta).

Cf. A000219, A001477, A014153, A095944, A161870.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*numtheory[sigma][2](j), j=1..n)/n)
    end:
a:= n-> add(b(n-j)*j, j=0..n):
seq(a(n), n=0..42);  # Alois P. Heinz, Feb 09 2023

nmax = 50; CoefficientList[Series[x/(1-x)^2 * Product[1/(1 - x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) = Sum_{k=0..n} A000219(k) * (n-k).
G.f.: x/(1-x)^2 * Product_{k>=1} 1/(1 - x^k)^k.
a(n) ~ exp(1/12 + 3*zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (A * sqrt(3*Pi) * 2^(35/36) * zeta(3)^(17/36) * n^(1/36)), where A is the Glaisher-Kinkelin constant A074962.

cp's OEIS Frontend

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

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A276551 Convolution square of A073592.

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A261384 Expansion of Product_{k>=1} (1+x^k)^(2*k-1) / (1-x^k)^(2*k).

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A321947 Column k=2 of triangle A257673.

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A383353 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where n 2-colored objects are distributed into k containers of two kinds. Containers may be left empty.

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A295828 Expansion of Product_{k>=1} 1/(1 - x^k)^(2*k*(2*k-1)).

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A360489 Convolution of A000219 and A001477.

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A261384 Expansion of Product_{k>=1} (1+x^k)^(2k-1) / (1-x^k)^(2k).

A295828 Expansion of Product_{k>=1} 1/(1 - x^k)^(2k(2*k-1)).