A160461 -id:A160461 - cp's OEIS frontend

A182821 Expansion of g.f.: exp( Sum_{n>=1} sigma(5n)x^n/n ).

1, 6, 27, 98, 315, 917, 2486, 6345, 15427, 35965, 80897, 176296, 373652, 772381, 1561130, 3091476, 6008896, 11480887, 21591830, 40016045, 73157052, 132052382, 235535752, 415433365, 725043875, 1252857043, 2144601961, 3638413830

Offset: 0

Paul D. Hanna, Dec 05 2010

sigma(5*n) = A000203(5*n), the sum of divisors of 5n.

Compare g.f. to P(x), the g.f. of partition numbers (A000041): P(x) = exp( Sum_{n>=1} sigma(n)*x^n/n ).

			G.f.: A(x) = 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 917*x^5 + ...
log(A(x)) = 6*x + 18*x^2/2 + 24*x^3/3 + 42*x^4/4 + 31*x^5/5 + 72*x^6/6 + 48*x^7/7 + 90*x^8/8 + ... + sigma(5n)*x^n/n + ...

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

Cf. A000203, A000041; variants: A182818, A182819, A182820.
Cf. Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^k: A035959 (k=1), A160461 (k=2), A277212 (k=5),
this sequence (k=6).

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^(5*j))/(1 - x^j)^6: j in [1..(m+2)]]) )); // G. C. Greubel, Nov 18 2018

nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))/(1 - x^k)^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2016 *)

{a(n)=polcoeff(exp(sum(m=1,n,sigma(5*m)*x^m/m)+x*O(x^n)),n)}

default(seriesprecision,66); Vec(eta(x^5)/eta(x)^6) \\ Joerg Arndt, Dec 05 2010

m=30; x='x+O('x^m); Vec(prod(j=1,m+2, (1 - x^(5*j))/(1 - x^j)^6)) \\ G. C. Greubel, Nov 18 2018

R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(30)
s = prod((1 - x^(5*j))/(1 - x^j)^6 for j in (1..32))
list(s) # G. C. Greubel, Nov 18 2018

G.f.: A(x) = E(x^5)/E(x)^6 where E(x) = Product_{k>=1} (1-x^k). - Joerg Arndt, Dec 05 2010
a(n) ~ 29^(3/2) * exp(sqrt(58*n/15)*Pi) / (2400*sqrt(3)*n^2). - Vaclav Kotesovec, Nov 28 2016
A(x^5) = P(x)*P(a*x)*P(a^2*x)*P(a^3*x)*P(a^4*x), where P(x) = 1/Product_{n>=1} (1 - x^n) is the g.f. for the partition function p(n) = A000041(n), and where a = exp(2*Pi*i/5) is a primitive fifth root of unity. - Peter Bala, Jan 24 2017

A160460 Coefficients in the expansion of C^6 / B^7, in Watson's notation of page 106.

Original entry on oeis.org

1, 7, 35, 140, 490, 1541, 4480, 12195, 31465, 77525, 183626, 420077, 932030, 2011905, 4237130, 8725671, 17605602, 34861815, 67848095, 129946805, 245203642, 456303872, 838178470, 1520969100, 2728472695, 4841909821, 8504898720, 14794863270, 25500965320

Offset: 0

N. J. A. Sloane, Nov 13 2009

nonn

			x^23 + 7*x^47 + 35*x^71 + 140*x^95 + 490*x^119 + 1541*x^143 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Watson, G. N., Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.

Cf. Product_{n>=1} (1 - x^(5*n))^k/(1 - x^n)^(k + 1): A160461 (k=1), A160462 (k=2), A160463 (k=3), A160506 (k=4), A071734 (k=5), this sequence (k=6), A160521 (k=7), A278555 (k=12), A278556 (k=18), A278557 (k=24), A278558 (k=30).

nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^6/(1 - x^k)^7, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2016 *)

See Maple code in A160458 for formula.

a(n) ~ sqrt(29/15) * exp(Pi*sqrt(58*n/15)) / (500*n). - Vaclav Kotesovec, Nov 28 2016

A278556 Expansion of Product_{n>=1} (1 - x^(5*n))^18/(1 - x^n)^19 in powers of x.

Original entry on oeis.org

1, 19, 209, 1710, 11495, 66862, 347339, 1645875, 7221520, 29668595, 115116233, 424720338, 1498263563, 5076482415, 16583497160, 52399330389, 160586833362, 478482249548, 1388989067820, 3935549005725, 10901608510397, 29565343541110, 78604103339462

Offset: 0

Seiichi Manyama, Nov 23 2016

nonn

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

Cf. Product_{n>=1} (1 - x^(5*n))^k/(1 - x^n)^(k + 1): A160461 (k=1), A160462 (k=2), A160463 (k=3), A160506 (k=4), A071734 (k=5), A160460 (k=6), A160521 (k=7), A278555 (k=12), this sequence (k=18), A278557 (k=24), A278558 (k=30).

CoefficientList[ Series[ Product[(1 - x^(5n))^18/(1 - x^n)^19, {n, 22}], {x, 0, 22}], x] (* Robert G. Wilson v, Nov 24 2016 *)

G.f.: Product_{n>=1} (1 - x^(5*n))^18/(1 - x^n)^19.
A278559(n) = 5^2*63*A160460(n) + 5^5*52*A278555(n-1) + 5^7*63*a(n-2) + 5^10*6*A278557(n-3) + 5^12*A278558(n-4) for n >= 4.
a(n) ~ sqrt(77/15) * exp(Pi*sqrt(154*n/15)) / (7812500*n). - Vaclav Kotesovec, Nov 28 2016

A160462 Coefficients in the expansion of C^2/B^3, in Watson's notation of page 106.

Original entry on oeis.org

1, 3, 9, 22, 51, 106, 215, 411, 766, 1377, 2423, 4154, 7001, 11567, 18834, 30195, 47809, 74735, 115585, 176847, 268064, 402598, 599695, 886116, 1299808, 1893115, 2739248, 3938491, 5629407, 8000431, 11309295, 15904003, 22256183, 30998479, 42981170, 59337604

Offset: 0

N. J. A. Sloane, Nov 13 2009

nonn

			x^7+3*x^31+9*x^55+22*x^79+51*x^103+106*x^127+215*x^151+...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Watson, G. N., Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.

Cf. Product_{n>=1} (1 - x^(5*n))^k/(1 - x^n)^(k + 1): A160461 (k=1), this sequence (k=2), A160463 (k=3), A160506 (k=4), A071734 (k=5), A160460 (k=6), A160521 (k=7), A278555 (k=12), A278556 (k=18), A278557 (k=24), A278558 (k=30).

nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^2/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2016 *)

See Maple code in A160458 for formula.

a(n) ~ sqrt(13/15) * exp(Pi*sqrt(26*n/15)) / (20*n). - Vaclav Kotesovec, Nov 28 2016

A160463 Coefficients in the expansion of C^3/B^4, in Watson's notation of page 106.

Original entry on oeis.org

1, 4, 14, 40, 105, 249, 562, 1198, 2460, 4865, 9352, 17486, 31973, 57220, 100550, 173665, 295413, 495339, 819900, 1340655, 2167825, 3468579, 5495908, 8628080, 13428945, 20730689, 31757174, 48293585, 72933885, 109421095, 163135433, 241763735, 356246552

Offset: 0

N. J. A. Sloane, Nov 13 2009

nonn

			x^11+4*x^35+14*x^59+40*x^83+105*x^107+249*x^131+562*x^155+...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Watson, G. N., Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.

Cf. Product_{n>=1} (1 - x^(5*n))^k/(1 - x^n)^(k + 1): A160461 (k=1), A160462 (k=2), this sequence (k=3), A160506 (k=4), A071734 (k=5), A160460 (k=6), A160521 (k=7), A278555 (k=12), A278556 (k=18), A278557 (k=24), A278558 (k=30).

nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^3/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2016 *)

See Maple code in A160458 for formula.

a(n) ~ sqrt(17/3) * exp(Pi*sqrt(34*n/15)) / (100*n). - Vaclav Kotesovec, Nov 28 2016

A160506 Coefficients in the expansion of C^4/B^5, in Watson's notation of page 106.

Original entry on oeis.org

1, 5, 20, 65, 190, 502, 1245, 2910, 6505, 13965, 29005, 58455, 114810, 220240, 413775, 762635, 1381550, 2463060, 4327445, 7500260, 12836645, 21712470, 36323930, 60143320, 98620425, 160238035, 258110955, 412367705, 653709340, 1028658150, 1607306688

Offset: 0

N. J. A. Sloane, Nov 13 2009

nonn

			x^15+5*x^39+20*x^63+65*x^87+190*x^111+502*x^135+1245*x^159+...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Watson, G. N., Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.

Cf. Product_{n>=1} (1 - x^(5*n))^k/(1 - x^n)^(k + 1): A160461 (k=1), A160462 (k=2), A160463 (k=3), this sequence (k=4), A071734 (k=5), A160460 (k=6), A160521 (k=7), A278555 (k=12), A278556 (k=18), A278557 (k=24), A278558 (k=30).

nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^4/(1 - x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2016 *)

See Maple code in A160458 for formula.

a(n) ~ sqrt(7/5) * exp(Pi*sqrt(14*n/5)) / (100*n). - Vaclav Kotesovec, Nov 28 2016

A160521 Coefficients in the expansion of C^7/B^8, in Watson's notation of page 106.

Original entry on oeis.org

1, 8, 44, 192, 726, 2457, 7648, 22220, 60993, 159478, 399906, 966600, 2261630, 5139897, 11378988, 24598683, 52033372, 107890610, 219630050, 439535138, 865784403, 1680352500, 3216454360, 6077280123, 11343018559, 20928404349, 38194869384, 68989715838

Offset: 0

N. J. A. Sloane, Nov 13 2009

nonn

			x^27+8*x^51+44*x^75+192*x^99+726*x^123+2457*x^147+7648*x^171+...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Watson, G. N., Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.

Cf. Product_{n>=1} (1 - x^(5*n))^k/(1 - x^n)^(k + 1): A160461 (k=1), A160462 (k=2), A160463 (k=3), A160506 (k=4), A071734 (k=5), A160460 (k=6), this sequence (k=7), A278555 (k=12), A278556 (k=18), A278557 (k=24), A278558 (k=30).

nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^7/(1 - x^k)^8, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2016 *)

See Maple code in A160458 for formula.

a(n) ~ sqrt(11) * exp(Pi*sqrt(22*n/5)) / (2500*n). - Vaclav Kotesovec, Nov 28 2016

A160525 Coefficients in the expansion of C/B^2, in Watson's notation of page 118.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 65, 109, 183, 295, 471, 732, 1129, 1705, 2554, 3769, 5517, 7979, 11458, 16289, 23007, 32227, 44869, 62028, 85284, 116530, 158432, 214228, 288348, 386224, 515156, 684109, 904963, 1192353, 1565383, 2047642, 2669591, 3468797, 4493351, 5802533

Offset: 0

N. J. A. Sloane, Nov 13 2009

nonn

			G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 20*x^4 + 36*x^5 + 65*x^6 + 109*x^7 + ...
G.f. = q^5 + 2*q^29 + 5*q^53 + 10*q^77 + 20*q^101 + 36*q^125 + 65*q^149 + 109*q^173 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.

Cf. A160526, A160527, A160528.

Cf. Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^2: A000041 (k=1), A015128 (k=2), A278690 (k=3), A160461 (k=5), this sequence (k=7).

M1:=1200:
fm:=mul(1-x^n,n=1..M1):
A:=x^(1/7)*subs(x=x^(24/7),fm):
B:=x*subs(x=x^24,fm):
C:=x^7*subs(x=x^168,fm):
t1:=C/B^2;
t2:=series(t1,x,M1);
t3:=subs(x=y^(1/24),t2/x^5);
t4:=series(t3,y,M1/24);
t5:=seriestolist(t4); # A160525

nmax = 50; CoefficientList[Series[Product[(1 - x^(7*k))/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 13 2017 *)

See Maple code for formula.
G.f.: Product_{n>=1} (1 - x^(7*n))/(1 - x^n)^2. - Seiichi Manyama, Nov 06 2016
a(n) ~ sqrt(13/3) * exp(sqrt(26*n/21)*Pi) / (28*n). - Vaclav Kotesovec, Apr 13 2017

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

Original entry on oeis.org

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

A278690 Expansion of Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^2 in powers of x.

Original entry on oeis.org

1, 2, 5, 9, 18, 31, 54, 88, 144, 225, 351, 531, 800, 1179, 1728, 2492, 3573, 5058, 7119, 9918, 13743, 18882, 25810, 35028, 47313, 63513, 84883, 112833, 149373, 196803, 258309, 337590, 439650, 570357, 737496, 950270, 1220688, 1563021, 1995642, 2540466, 3225386

Offset: 0

Seiichi Manyama, Nov 26 2016

			G.f. = 1 + 2*x + 5*x^2 + 9*x^3 + 18*x^4 + 31*x^5 + 54*x^6 + ...
G.f. = q + 2*q^25 + 5*q^49 + 9*q^73 + 18*q^97 + 31*q^121 + 54*q^145 + ... - _Michael Somos_, Nov 25 2019

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

Cf. Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^k: A000726 (k=1), this sequence (k=2), A273845 (k=3), A182819 (k=4).
Cf. Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^2: A000041 (k=1), A015128 (k=2), this sequence (k=3), A160461 (k=5).
Cf. A298311.

nmax = 50; CoefficientList[Series[Product[(1 - x^(3*k))/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 26 2016 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3] / QPochhammer[ x]^2, {x, 0, n}]; (* Michael Somos, Nov 25 2019 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) / eta(x + A)^2, n))}; /* Michael Somos, Nov 25 2019 */

G.f.: Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^2.
a(n) ~ sqrt(5/3)*exp(sqrt(10*n)*Pi/3)/(12*n). - Vaclav Kotesovec, Nov 26 2016
Expansion of q^(-1/24) * eta(q^3) / eta(q)^2 in powers of q. - Michael Somos, Nov 25 2019
G.f.: 1/Product_{n > = 1} ( 1 - x^(n/gcd(n,k)) ) for k = 3. Cf. A000041 (k = 1), A015128 (k = 2), A298311 (k = 4) and A160461 (k = 5). - Peter Bala, Nov 17 2020

cp's OEIS Frontend

A182821 Expansion of g.f.: exp( Sum_{n>=1} sigma(5*n)*x^n/n ).

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A160460 Coefficients in the expansion of C^6 / B^7, in Watson's notation of page 106.

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A278556 Expansion of Product_{n>=1} (1 - x^(5*n))^18/(1 - x^n)^19 in powers of x.

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A160462 Coefficients in the expansion of C^2/B^3, in Watson's notation of page 106.

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A160463 Coefficients in the expansion of C^3/B^4, in Watson's notation of page 106.

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A160506 Coefficients in the expansion of C^4/B^5, in Watson's notation of page 106.

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A160521 Coefficients in the expansion of C^7/B^8, in Watson's notation of page 106.

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A160525 Coefficients in the expansion of C/B^2, in Watson's notation of page 118.

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A182821 Expansion of g.f.: exp( Sum_{n>=1} sigma(5n)x^n/n ).

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