A278710 -id:A278710 - cp's OEIS frontend

A255528 G.f.: Product_{k>=1} 1/(1+x^k)^k.

1, -1, -1, -2, 1, 0, 4, 2, 8, -2, 4, -11, -1, -25, -5, -35, 13, -26, 49, -6, 110, 6, 159, -23, 182, -141, 129, -358, 62, -640, 39, -897, 237, -1013, 771, -914, 1793, -664, 3143, -565, 4635, -1157, 5727, -3119, 6121, -7041, 5642, -13088, 5097, -20758, 5879

Offset: 0

Vaclav Kotesovec, Feb 24 2015

sign

In general, if m >= 1 and g.f. = Product_{k>=1} 1/(1 + x^k)^(m*k), then a(n, m) ~ (-1)^n * exp(-m/12 + 3 * 2^(-5/3) * m^(1/3) * Zeta(3)^(1/3) * n^(2/3)) * 2^(m/18 - 5/6) * A^m * m^(1/6 - m/36) * Zeta(3)^(1/6 - m/36) * n^(m/36 - 2/3) / sqrt(3*Pi), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 13 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Vaclav Kotesovec)
Vaclav Kotesovec, Graph - the asymptotic ratio (250000 terms)
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 20.

Cf. A000219, A026007, A026011, A073592, A262736.

Cf. A278710 (m=2), A279031 (m=3), A279411 (m=4), A279932 (m=5).

with(numtheory): A000219:=proc(n) option remember; if n = 0 then 1 else add(sigma[2](k)*A000219(n-k), k = 1..n)/n fi: end: A073592:=proc(n) option remember; if n = 0 then 1 else -add(sigma[2](k)*A073592(n-k), k = 1..n)/n fi: end: a:=proc(n); add(A073592(n-2*m)*A000219(m), m = 0..floor(n/2)): end: seq(a(n), n = 0..50); # Vaclav Kotesovec, Mar 09 2015

nmax=100; CoefficientList[Series[Product[1/(1+x^k)^k,{k,1,nmax}],{x,0,nmax}],x]

{a(n) = if(n<0, 0, polcoeff(exp(sum(k=1, n, (-1)^k * x^k / (1-x^k)^2 / k, x*O(x^n))), n))}
for(n=0, 100, print1(a(n), ", "))

a(n) ~ (-1)^n * A * Zeta(3)^(5/36) * exp(3*Zeta(3)^(1/3)*n^(2/3)/2^(5/3) - 1/12) / (2^(7/9) * sqrt(3*Pi) * n^(23/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Sep 29 2015

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A078306(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017

A279031 Expansion of Product_{k>0} 1/(1 + x^k)^(k*3).

Original entry on oeis.org

1, -3, 0, -1, 15, -3, 8, -42, 6, -83, 81, -39, 316, -90, 420, -603, 363, -1656, 625, -2556, 2877, -2599, 7818, -3483, 13886, -11049, 17040, -31493, 20196, -63876, 39244, -96453, 105891, -120431, 243333, -164100, 440873, -327387, 643968, -765115, 840207

Offset: 0

Seiichi Manyama, Apr 11 2017

sign

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

Product_{k>0} 1/(1 + x^k)^(k*m): A027346 (m=-3), A255528 (m=1), A278710 (m=2), this sequence (m=3), A279411 (m=4).

a(n) ~ (-1)^n * exp(-1/4 + 2^(-5/3) * 3^(4/3) * Zeta(3)^(1/3) * n^(2/3)) * A^3 * Zeta(3)^(1/12) / (2^(2/3) * 3^(5/12) * sqrt(Pi) * n^(7/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 13 2017

G.f.: exp(3*Sum_{k>=1} (-1)^k*x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, Mar 27 2018

A279411 Expansion of Product_{k>0} 1/(1 + x^k)^(k*4).

Original entry on oeis.org

1, -4, 2, 0, 23, -20, 2, -88, 63, -96, 318, -104, 626, -844, 504, -2472, 1525, -3704, 6184, -4288, 15284, -10736, 23254, -35792, 30228, -84544, 60974, -139240, 176658, -190108, 418940, -320976, 755332, -773524, 1111678, -1847304, 1669046, -3634296

Offset: 0

Seiichi Manyama, Apr 11 2017

sign

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

Column k=4 of A279928.

Product_{k>0} 1/(1 + x^k)^(k*m): A027906 (m=-4), A255528 (m=1), A278710 (m=2), A279031 (m=3), this sequence (m=4), A279932 (m=5).

a(n) ~ (-1)^n * exp(-1/3 + 3/2 * Zeta(3)^(1/3) * n^(2/3)) * A^4 * Zeta(3)^(1/18) / (sqrt(6*Pi) * n^(5/9)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 13 2017

G.f.: exp(4*Sum_{k>=1} (-1)^k*x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, Mar 27 2018

A279928 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1+x^j)^(j*k) in powers of x.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, -1, 0, 1, -3, -1, -2, 0, 1, -4, 0, -2, 1, 0, 1, -5, 2, -1, 7, 0, 0, 1, -6, 5, 0, 15, 2, 4, 0, 1, -7, 9, 0, 23, -3, 10, 2, 0, 1, -8, 14, -2, 30, -20, 8, -8, 8, 0, 1, -9, 20, -7, 36, -51, 2, -42, 5, -2, 0, 1, -10, 27, -16, 42, -96, 5, -88, 6

Offset: 0

Seiichi Manyama, Apr 11 2017

			Square array begins:
   1,  1,  1,  1,  1, ...
   0, -1, -2, -3, -4, ...
   0, -1, -1,  0,  2, ...
   0, -2, -2, -1,  0, ...
   0,  1,  7, 15, 23, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-5 give: A000007, A255528, A278710, A279031, A279411, A279932.
Main diagonal gives A281266.
Antidiagonal sums give A299212.
Cf. A255961, A277938.

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

A279932 Expansion of Product_{k>0} 1/(1 + x^k)^(k*5).

Original entry on oeis.org

1, -5, 5, 0, 30, -51, 5, -130, 220, -125, 649, -605, 870, -2695, 1565, -4852, 7915, -6360, 20625, -17880, 33551, -61015, 50865, -138510, 135485, -224725, 389025, -359610, 849525, -838970, 1417404, -2195205, 2275690, -4756040, 4657940, -8315123, 11174840, -13352315

Offset: 0

Seiichi Manyama, Apr 12 2017

sign

In general, if m >= 1 and g.f. = Product_{k>=1} 1/(1 + x^k)^(m*k), then a(n, m) ~ (-1)^n * exp(-m/12 + 3 * 2^(-5/3) * m^(1/3) * Zeta(3)^(1/3) * n^(2/3)) * 2^(m/18 - 5/6) * A^m * m^(1/6 - m/36) * Zeta(3)^(1/6 - m/36) * n^(m/36 - 2/3) / sqrt(3*Pi), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 13 2017

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

Column k=5 of A279928.

Product_{k>0} 1/(1 + x^k)^(k*m): A027906 (m=-4), A255528 (m=1), A278710 (m=2), A279031 (m=3), A279411 (m=4), this sequence (m=5).

a(n) ~ (-1)^n * exp(-5/12 + 3 * 2^(-5/3) * (5*Zeta(3))^(1/3) * n^(2/3)) * A^5 * (5*Zeta(3))^(1/36) / (2^(5/9) * sqrt(3*Pi) * n^(19/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 13 2017

G.f.: exp(5*Sum_{k>=1} (-1)^k*x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, Mar 27 2018

A284467 Expansion of Product_{k>=1} (1 + x^(2k-1))^(2k-1)/(1 + x^(2k))^(2k).

Original entry on oeis.org

1, 1, -2, 1, 2, -2, 0, -5, 10, 1, -15, 10, -1, 18, -39, 4, 50, -24, -14, -69, 165, -70, -83, -20, 154, 161, -550, 313, 55, 410, -960, 102, 1074, -406, -506, -1344, 3581, -1791, -833, -1833, 4995, 205, -6993, 2982, 2461, 7649, -19791, 9495, 4986, 9581, -26745, 0

Offset: 0

Seiichi Manyama, Apr 15 2017

sign

Robert Israel, Table of n, a(n) for n = 0..2000

Cf. A224364, A262736, A278710, A281683, A284474.

N:= 100: # to get a(0)..a(N)
P:= mul((1+x^(2*k-1))^(2*k-1)/(1+x^(2*k))^(2*k),k=1..N/2):
S:= series(P,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Apr 16 2017

nmax = 60; CoefficientList[Series[Product[(1 + x^(2*k-1))^(2*k-1)/(1 + x^(2*k))^(2*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 + x^k)^2)). - Ilya Gutkovskiy, Jun 20 2018

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

cp's OEIS Frontend

A255528 G.f.: Product_{k>=1} 1/(1+x^k)^k.

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A279031 Expansion of Product_{k>0} 1/(1 + x^k)^(k*3).

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A279411 Expansion of Product_{k>0} 1/(1 + x^k)^(k*4).

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A279928 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1+x^j)^(j*k) in powers of x.

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A279932 Expansion of Product_{k>0} 1/(1 + x^k)^(k*5).

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A284467 Expansion of Product_{k>=1} (1 + x^(2*k-1))^(2*k-1)/(1 + x^(2*k))^(2*k).

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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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A284467 Expansion of Product_{k>=1} (1 + x^(2k-1))^(2k-1)/(1 + x^(2k))^(2k).