A015128 -id:A015128 - cp's OEIS frontend

A298311 Expansion of Product_{k>=1} 1/((1 - x^(2k))(1 - x^(2*k-1))^3).

1, 3, 7, 16, 32, 61, 112, 197, 336, 560, 912, 1456, 2287, 3536, 5392, 8123, 12096, 17824, 26016, 37632, 53984, 76848, 108601, 152432, 212592, 294704, 406201, 556864, 759488, 1030784, 1392496, 1872784, 2508048, 3345184, 4444384, 5882747, 7758736, 10197712, 13358944, 17444256, 22708719

Offset: 0

Ilya Gutkovskiy, Jan 17 2018

Number of partitions of n where there are 3 kinds of odd parts.

Convolution of the sequences A000009 and A015128.

Michael D. Hirschhorn and James A. Sellers, A Family of Congruences Modulo 7 for Partitions with Monochromatic Even Parts and Multi-Colored Odd Parts, arXiv:2507.09752 [math.CO], 2025. See p. 2.
Index entries for related partition-counting sequences

Cf. A000009, A000041, A000716, A015128, A029862, A029863, A160461, A182818, A278690.

nmax = 40; CoefficientList[Series[Product[1/((1 - x^(2 k)) (1 - x^(2 k - 1))^3), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Product[(1 + x^k)^2/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/((1 - x^(2*k))*(1 - x^(2*k-1))^3).
G.f.: Product_{k>=1} (1 + x^k)^2/(1 - x^k).
a(n) ~ exp(2*Pi*sqrt(n/3)) / (2^(5/2)*sqrt(3)*n). - Vaclav Kotesovec, Apr 08 2018
G.f.: 1/Product_{n > = 1} ( 1 - x^(n/gcd(n,k)) ) for k = 4. Cf. A000041 (k = 1), A015128 (k = 2), A278690 (k = 3) and A160461 (k = 5). - Peter Bala, Nov 17 2020

A300415 Expansion of Product_{k>=2} (1 + x^k)/(1 - x^k).

Original entry on oeis.org

1, 0, 2, 2, 4, 6, 10, 14, 22, 32, 46, 66, 94, 130, 182, 250, 340, 462, 622, 830, 1106, 1462, 1922, 2518, 3282, 4256, 5502, 7082, 9078, 11602, 14774, 18746, 23722, 29922, 37630, 47202, 59044, 73662, 91682, 113830, 140994, 174262, 214906, 264462, 324802, 398110, 487018, 594694

Offset: 0

Ilya Gutkovskiy, Mar 05 2018

nonn

Convolution of the sequences A002865 and A025147.

Also number of overpartitions of n without a 1. - George Beck, Jan 25 2021

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for related partition-counting sequences

Cf. A002865, A015128, A025147, A207641, A211971.

g:= (1-x)/((1+x)*JacobiTheta4(0,x)):
S:=series(g,x,101):
seq(coeff(S,x,j),j=0..100); # Robert Israel, Mar 05 2018

nmax = 47; CoefficientList[Series[Product[(1 + x^k)/(1 - x^k), {k, 2, nmax}], {x, 0, nmax}], x]
nmax = 47; CoefficientList[Series[(1 - x)/((1 + x) EllipticTheta[4, 0, x]), {x, 0, nmax}], x]

G.f.: Product_{k>=2} (1 + x^k)/(1 - x^k).
G.f.: (1 - x)/((1 + x)*theta_4(x)), where theta_4() is the Jacobi theta function.
a(n) ~ Pi * exp(Pi*sqrt(n)) / (32*n^(3/2)). - Vaclav Kotesovec, Mar 05 2018

A303360 Expansion of Product_{n>=1} ((1 + 4x^n)/(1 - 4x^n))^(1/4).

Original entry on oeis.org

1, 2, 4, 18, 34, 166, 384, 1902, 4756, 24022, 64284, 321542, 899658, 4455690, 12888944, 63185250, 187513426, 910880550, 2759413788, 13295839638, 40967821494, 195979968882, 612569599440, 2911592648458, 9213101043936, 43538337410474, 139246245625364

Offset: 0

Seiichi Manyama, Apr 22 2018

nonn

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

Expansion of Product_{n>=1} ((1 + 2^b*x^n)/(1 - 2^b*x^n))^(1/(2^b)): A015128 (b=0), A303346 (b=1), this sequence (b=2).

Cf. A303361, A303391, A067855, A303350, A303392.

seq(coeff(series(mul(((1+4*x^k)/(1-4*x^k))^(1/4), k = 1..n), x, n+1), x, n), n = 0..35); # Muniru A Asiru, Apr 22 2018

nmax = 30; CoefficientList[Series[Product[((1 + 4*x^k)/(1 - 4*x^k))^(1/4), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 22 2018 *)
nmax = 30; CoefficientList[Series[(-3*QPochhammer[-4, x] / (5*QPochhammer[4, x]))^(1/4), {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 23 2018 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, ((1+4*x^k)/(1-4*x^k))^(1/4)))

a(n) ~ c * 4^n / n^(3/4), where c = (QPochhammer[-1, 1/4] / QPochhammer[1/4])^(1/4) / Gamma(1/4) = 0.3885547372628... - Vaclav Kotesovec, Apr 23 2018

A303361 Expansion of Product_{n>=1} ((1 + (4x)^n)/(1 - (4x)^n))^(1/4).

Original entry on oeis.org

1, 2, 10, 60, 262, 1372, 7044, 32760, 153670, 789676, 3659820, 17109320, 83073180, 381273240, 1786996424, 8604391920, 38832248902, 179714213580, 845485079580, 3834271942440, 17666638985652, 81920437065288, 370224975781560, 1685489994025360

Offset: 0

Seiichi Manyama, Apr 22 2018

nonn

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

Expansion of Product_{n>=1} ((1 + (2^b*x)^n)/(1 - (2^b)*x^n))^(1/(2^b)): A015128 (b=0), A303307 (b=1), this sequence (b=2).

Cf. A303360.

seq(coeff(series(mul(((1+(4*x)^k)/(1-(4*x)^k))^(1/4), k = 1..n), x, n+1), x, n), n = 0..35); # Muniru A Asiru, Apr 22 2018

nmax = 30; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(1/4), {k, 1, nmax}], {x, 0, nmax}], x] * 4^Range[0, nmax] (* Vaclav Kotesovec, Apr 23 2018 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, ((1+(4*x)^k)/(1-(4*x)^k))^(1/4)))

a(n) ~ 2^(2*n - 5/2) * exp(sqrt(n)*Pi/2) / n^(13/16). - Vaclav Kotesovec, Apr 23 2018

A321884 Number A(n,k) of partitions of n into colored blocks of equal parts with colors from a set of size k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 4, 3, 0, 1, 4, 6, 8, 5, 0, 1, 5, 8, 15, 14, 7, 0, 1, 6, 10, 24, 27, 24, 11, 0, 1, 7, 12, 35, 44, 51, 40, 15, 0, 1, 8, 14, 48, 65, 88, 93, 64, 22, 0, 1, 9, 16, 63, 90, 135, 176, 159, 100, 30, 0, 1, 10, 18, 80, 119, 192, 295, 312, 264, 154, 42, 0

Offset: 0

Alois P. Heinz, Aug 27 2019

			A(3,2) = 8: 3a, 3b, 2a1a, 2a1b, 2b1a, 2b1b, 111a, 111b.
Square array A(n,k) begins:
  1,  1,   1,   1,   1,   1,    1,    1,    1, ...
  0,  1,   2,   3,   4,   5,    6,    7,    8, ...
  0,  2,   4,   6,   8,  10,   12,   14,   16, ...
  0,  3,   8,  15,  24,  35,   48,   63,   80, ...
  0,  5,  14,  27,  44,  65,   90,  119,  152, ...
  0,  7,  24,  51,  88, 135,  192,  259,  336, ...
  0, 11,  40,  93, 176, 295,  456,  665,  928, ...
  0, 15,  64, 159, 312, 535,  840, 1239, 1744, ...
  0, 22, 100, 264, 544, 970, 1572, 2380, 3424, ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened
Jessica Jay and Benjamin Lees, Combinatorial identities from an inhomogeneous Ising chain, arXiv:2401.16311 [math.PR], 2024.
Wikipedia, Partition (number theory)

Columns k=0-4 give: A000007, A000041, A015128, A264686, A266821.
Main diagonal gives A321880.
Cf, A000142, A003056, A321878.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t, b[t, Min[t, i - 1], k]][n - i j], {j, 1, n/i}] k + b[n, i - 1, k]]];
A[n_, k_] := b[n, n, k];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *)

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

A(n,k) = Sum_{i=0..floor((sqrt(1+8*k)-1)/2)} k!/(k-i)! * A321878(n,i).

A358369 Euler transform of 2^floor(n/2), (A016116).

Original entry on oeis.org

1, 1, 3, 5, 12, 20, 43, 73, 146, 250, 475, 813, 1499, 2555, 4592, 7800, 13761, 23253, 40421, 67963, 116723, 195291, 332026, 552882, 932023, 1544943, 2585243, 4267081, 7094593, 11662769, 19281018, 31575874, 51937608, 84753396, 138772038, 225693778, 368017636

Offset: 0

Peter Luschny, Nov 17 2022

nonn

Cf. A002513, A016116.

Sequences that can be represented as a EulerTransform(BinaryRecurrenceSequence()) include A000009, A000041, A000712, A001970, A002513, A010054, A015128, A022567, A034691, A111317, A111335, A117410, A156224, A166861, A200544, A261031, A261329, A358449.

BinaryRecurrenceSequence := proc(b, c, u0:=0, u1:=1) local u;
u := proc(n) option remember; if n < 2 then return [u0, u1][n + 1] fi;
b*u(n - 1) + c*u(n - 2) end; u end:
EulerTransform := proc(a) local b;
b := proc(n) option remember; if n = 0 then return 1 fi; add(add(d * a(d),
d = NumberTheory:-Divisors(j)) * b(n-j), j = 1..n) / n end; b end:
a := EulerTransform(BinaryRecurrenceSequence(0, 2, 1)): seq(a(n), n=0..36);

from typing import Callable
from functools import cache
from sympy import divisors
def BinaryRecurrenceSequence(b:int, c:int, u0:int=0, u1:int=1) -> Callable:
    @cache
    def u(n: int) -> int:
        if n < 2:
            return [u0, u1][n]
        return b * u(n - 1) + c * u(n - 2)
    return u
def EulerTransform(a: Callable) -> Callable:
    @cache
    def b(n: int) -> int:
        if n == 0:
            return 1
        s = sum(sum(d * a(d) for d in divisors(j)) * b(n - j)
            for j in range(1, n + 1))
        return s // n
    return b
b = BinaryRecurrenceSequence(0, 2, 1)
a = EulerTransform(b)
print([a(n) for n in range(37)])

# uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 2, 1)
a = EulerTransform(b)
print([a(n) for n in range(37)])

A274327 Expansion of Product_{n>=1} (1 - x^(4*n))/(1 - x^n)^4 in powers of x.

Original entry on oeis.org

1, 4, 14, 40, 104, 248, 560, 1200, 2474, 4924, 9520, 17928, 33008, 59528, 105408, 183536, 314744, 532208, 888382, 1465208, 2389808, 3857456, 6166096, 9766576, 15336816, 23888844, 36924656, 56659296, 86341664, 130710104, 196640576, 294059872, 437232746, 646561792

Offset: 0

Seiichi Manyama, Nov 07 2016

nonn

			G.f.: 1 + 4*x + 14*x^2 + 40*x^3 + 104*x^4 + 248*x^5 + 560*x^6 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol

Cf. Expansion of Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k in powers of x: A015128 (k=2), A273845 (k=3), this sequence (k=4), A277212 (k=5), A277283 (k=6), A160539 (k=7).

Cf. A083703.

nmax = 50; CoefficientList[Series[Product[(1 - x^(4*k))/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *)
(QPochhammer[x^4, x^4]/QPochhammer[x, x]^4 + O[x]^40)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)

first(n)=my(x='x);Vec(prod(k=1,n,(1-x^(4*k))/(1-x^k)^4,1+O(x^(n+1)))) \\ Charles R Greathouse IV, Nov 07 2016

G.f.: Product_{n>=1} (1 - x^(4*n))/(1 - x^n)^4.
a(n) ~ 5*exp(Pi*sqrt(5*n/2)) / (2^(13/2) * n^(3/2)). - Vaclav Kotesovec, Nov 10 2016
G.f.: (x^4; x^4)inf/((x; x)_inf)^4, where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2016
a(0) = 1, a(n) = (4/n)*Sum_{k=1..n} A285895(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017

A300274 G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = Product_{n>=1} (1 + x^n)/(1 - x^n).

Original entry on oeis.org

2, 2, 6, 10, 22, 30, 62, 86, 146, 206, 342, 454, 726, 974, 1442, 1962, 2862, 3762, 5398, 7094, 9834, 12942, 17726, 22938, 31042, 40094, 53254, 68518, 90246, 114914, 150142, 190550, 245906, 310942, 398554, 500474, 637590, 797534, 1007714, 1255850, 1578526, 1956786

Offset: 1

Ilya Gutkovskiy, Mar 01 2018

nonn

Moebius transform of A015128.

N. J. A. Sloane, Transforms

Cf. A000837, A015128, A078374, A300275, A300276, A300277, A300278.

nn = 42; f[x_] := 1 + Sum[a[n] x^n/(1 - x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - Product[(1 + x^n)/(1 - x^n), {n, 1, nn}], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
s[n_] := SeriesCoefficient[Product[(1 + x^k)/(1 - x^k), {k, 1, n}], {x, 0, n}]; a[n_] := Sum[MoebiusMu[n/d] s[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 42}]

a(n) = Sum_{d|n} mu(n/d)*A015128(d).

A303381 Expansion of Product_{n>=1} ((1 + (8x)^n)/(1 - (8x)^n))^(1/8).

Original entry on oeis.org

1, 2, 18, 204, 1526, 15228, 146676, 1217880, 10322982, 106429420, 886934236, 7632390312, 72137002428, 600860144728, 5351962341672, 51402944345520, 411439139563526, 3624067316629836, 33666668386023244, 279519776297893512, 2480351338204454484

Offset: 0

Seiichi Manyama, Apr 22 2018

nonn

In general, if h>=1 and g.f. = Product_{k>=1} ((1 + (h*x)^k)/(1 - (h*x)^k))^(1/h), then a(n) ~ h^n * exp(Pi*sqrt(n/h)) /(2^(3/2 + 3/(2*h)) * h^(1/4 + 1/(4*h)) * n^(3/4 + 1/(4*h))). - Vaclav Kotesovec, Apr 23 2018

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

Expansion of Product_{n>=1} ((1 + (2^b*x)^n)/(1 - (2^b)*x^n))^(1/(2^b)): A015128 (b=0), A303307 (b=1), A303361 (b=2), this sequence (b=3).

Cf. A303382.

seq(coeff(series(mul(((1+(8*x)^k)/(1-(8*x)^k))^(1/8), k = 1..n), x, n+1), x, n), n = 0..25); # Muniru A Asiru, Apr 23 2018

nmax = 20; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(1/8), {k, 1, nmax}], {x, 0, nmax}], x] * 8^Range[0, nmax] (* Vaclav Kotesovec, Apr 23 2018 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, ((1+(8*x)^k)/(1-(8*x)^k))^(1/8)))

a(n) ~ 2^(3*n - 81/32) * exp(sqrt(n)*Pi/2^(3/2)) / n^(25/32). - Vaclav Kotesovec, Apr 23 2018

A303391 Expansion of Product_{k>=1} (1 + 4x^k)/(1 - 4x^k).

Original entry on oeis.org

1, 8, 40, 200, 872, 3720, 15400, 62920, 254440, 1024648, 4112680, 16483400, 66000360, 264150920, 1056903080, 4228272200, 16914393832, 67660396040, 270647139240, 1082600410440, 4330424811880, 17321748357640, 69287088965800, 277148557003720, 1108594618342760

Offset: 0

Vaclav Kotesovec, Apr 23 2018

nonn

Robert Israel, Table of n, a(n) for n = 0..1600

Cf. A261568, A246936, A067855, A303350.

Cf. A015128, A261584, A303390.

N:= 50: # for a(0)..a(N)
G:= mul((1+4*x^k)/(1-4*x^k),k=1..N):
S:= series(G,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Feb 13 2019

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

a(n) ~ c * 4^n, where c = QPochhammer[-1, 1/4] / QPochhammer[1/4] = 3.9385207073365388638943873939345313401323799...

cp's OEIS Frontend

A298311 Expansion of Product_{k>=1} 1/((1 - x^(2*k))*(1 - x^(2*k-1))^3).

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A300415 Expansion of Product_{k>=2} (1 + x^k)/(1 - x^k).

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A303360 Expansion of Product_{n>=1} ((1 + 4*x^n)/(1 - 4*x^n))^(1/4).

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A303361 Expansion of Product_{n>=1} ((1 + (4*x)^n)/(1 - (4*x)^n))^(1/4).

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A321884 Number A(n,k) of partitions of n into colored blocks of equal parts with colors from a set of size k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A358369 Euler transform of 2^floor(n/2), (A016116).

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A274327 Expansion of Product_{n>=1} (1 - x^(4*n))/(1 - x^n)^4 in powers of x.

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A300274 G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = Product_{n>=1} (1 + x^n)/(1 - x^n).

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A298311 Expansion of Product_{k>=1} 1/((1 - x^(2k))(1 - x^(2*k-1))^3).

A303360 Expansion of Product_{n>=1} ((1 + 4x^n)/(1 - 4x^n))^(1/4).

A303361 Expansion of Product_{n>=1} ((1 + (4x)^n)/(1 - (4x)^n))^(1/4).

A303381 Expansion of Product_{n>=1} ((1 + (8x)^n)/(1 - (8x)^n))^(1/8).

A303391 Expansion of Product_{k>=1} (1 + 4x^k)/(1 - 4x^k).