A277212 -id:A277212 - 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

A109064 Expansion of eta(q)^5 / eta(q^5) in powers of q.

Original entry on oeis.org

1, -5, 5, 10, -15, -5, -10, 30, 25, -35, 5, -60, 30, 60, -30, 10, -55, 80, 35, -100, -15, -60, 60, 110, -50, -5, -60, 100, 90, -150, -10, -160, 105, 120, -80, 30, -105, 180, 100, -120, 25, -210, 60, 210, -180, -35, -110, 230, 110, -215, 5, -160, 180, 260

Offset: 0

Michael Somos, Jun 17 2005

sign

Number 12 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 - 5*q + 5*q^2 + 10*q^3 - 15*q^4 - 5*q^5 - 10*q^6 + 30*q^7 + 25*q^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 37.
William Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162. See page 151.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.
G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128. See the expression B^5/C in the notation of p. 106. [Added by _N. J. A. Sloane_, Nov 13 2009]

Cf. A053723, A109091, A138506, A277212, A285896.

A := Basis( ModularForms( Gamma1(5), 2), 54); A[1] - 5*A[2] + 5*A[3]; /* Michael Somos, May 19 2015 */

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d, 5)=0, -4, -5), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 07 2017

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^5 / QPochhammer[ q^5], {q, 0, n}]; (* Michael Somos, May 19 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], -5 DivisorSum[ n, # KroneckerSymbol[ 5, #] &]]; (* Michael Somos, May 19 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^5 / eta(x^5 + A), n))};

{a(n) = if( n<1, n==0, -5 * sumdiv(n, d, d * kronecker(5, d)))}; /* Michael Somos, May 19 2015 */

Euler transform of period 5 sequence [ -5, -5, -5, -5, -4, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 + 2 * u*v*w + u^2*w - 4 * u*w^2.
a(n) = -5 * b(n) where b() is multiplicative with a(0) = 1, b(p^e) = 1 if p=5,  b(p^e) = b(p) * b(p^(e-1)) - Kronecker(5, p) * p * b(p^(e-2)) otherwise. - Michael Somos, May 19 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (5 t)) = 5^(5/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A053723. - Michael Somos, May 19 2015
G.f.: Product_{k>0} (1 - x^k)^5 / (1 - x^(5*k)).
a(n) = -5 * A109091(n), unless n=0. a(n) = (-1)^n * A138506(n). a(5*n) = a(n).
a(0) = 1, a(n) = -(5/n)*Sum_{k=1..n} A285896(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(3*sqrt(5)) = 1.471273... . - Amiram Eldar, Jan 29 2024

A160458 Coefficients in the expansion of C^2/B^10, in Watson's notation of page 106.

Original entry on oeis.org

1, 10, 65, 330, 1430, 5510, 19395, 63440, 195250, 570570, 1594315, 4283270, 11113440, 27949580, 68340360, 162880080, 379227010, 864153940, 1930443705, 4233724000, 9127235430, 19364099520, 40470110005, 83395632580, 169581447000, 340533848010

Offset: 0

N. J. A. Sloane, Nov 13 2009

nonn

			G.f.: 1+10*q^24+65*q^48+330*q^72+1430*q^96+5510*q^120+19395*q^144+...

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. See the expression C^2/B^10.

Cf. A160459, A277212.

read format;
M1:=1200:
fm:=mul(1-x^n,n=1..M1):
A:=x^(1/5)*subs(x=x^(24/5),fm):
B:=x*subs(x=x^24,fm):
C:=x^5*subs(x=x^120,fm):
t1:=C^2/B^10;
t2:=series(t1,x,M1);
t3:=subs(x=y^(1/24),t2);
t4:=series(t3,y,M1/24);
t5:=seriestolist(t4); # A160458

x='x+O('x^66); Vec((eta(x^5)/eta(x)^5)^2) \\ Joerg Arndt, Nov 27 2016

See Maple code for formula.

a(n) = Sum_{k=0..n} A277212(k)*A277212(n-k). - Seiichi Manyama, Nov 27 2016

A273845 Expansion of Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^3 in powers of x.

Original entry on oeis.org

1, 3, 9, 21, 48, 99, 198, 375, 693, 1236, 2160, 3681, 6168, 10140, 16434, 26235, 41376, 64449, 99342, 151530, 229032, 343068, 509760, 751509, 1099998, 1598925, 2309274, 3314541, 4729920, 6711993, 9474624, 13306506, 18598437, 25874460, 35838288, 49427640, 67892592

Offset: 0

Seiichi Manyama, Nov 07 2016

nonn

			G.f.: 1 + 3*x + 9*x^2 + 21*x^3 + 48*x^4 + 99*x^5 + 198*x^6 + ...

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

Expansion of Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k in powers of x: A015128 (k=2), this sequence (k=3), A274327 (k=4), A277212 (k=5), A277283 (k=6), A160539 (k=7).

Cf. A005928, A132972.

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

first(n)=my(x='x); Vec(prod(k=1, n, (1-x^(3*k))/(1-x^k)^3, 1+O(x^(n+1)))) \\ Charles R Greathouse IV, Nov 07 2016

lista(nn) = {q='q+O('q^nn); Vec(eta(q^3)/eta(q)^3)} \\ Altug Alkan, Mar 20 2018

G.f.: Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^3.
a(n) ~ exp(4*Pi*sqrt(n)/3) / (9*sqrt(2)*n^(5/4)). - Vaclav Kotesovec, Nov 10 2016
G.f.: (x^3; x^3)inf/((x; x)_inf)^3, where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2016
a(0) = 1, a(n) = (3/n)*Sum_{k=1..n} A078708(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017
It appears that the g.f. A(x) = F(x)^3, where F(x) = exp( Sum_{n >= 0} x^(3*n+1)/((3*n + 1)*(1 - x^(3*n+1))) + x^(3*n+2)/((3*n + 2)*(1 - x^(3*n + 2))) ). Cf. A132972.  - Peter Bala, Dec 23 2021

A277974 Expansion of ((Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^5) - 1)/5 in powers of x.

Original entry on oeis.org

0, 1, 4, 13, 38, 101, 252, 594, 1340, 2907, 6104, 12447, 24744, 48068, 91476, 170838, 313646, 566824, 1009628, 1774290, 3079338, 5282172, 8962288, 15050848, 25032420, 41255101, 67406472, 109236685, 175654072, 280371510, 444372452, 699579062, 1094289564

Offset: 0

Seiichi Manyama, Nov 07 2016

nonn

			G.f. = x + 4*x^2 + 13*x^3 + 38*x^4 + 101*x^5 + 252*x^6 + ...

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

Cf. Expansion of ((Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k) - 1)/k in powers of x: A014968 (k=2), A277968 (k=3), this sequence (k=5), A160549 (k=7), A277912 (k=11).

nmax = 50; CoefficientList[Series[(Product[(1 - x^(5*j))/(1 - x^j)^5, {j, 1, nmax}] - 1)/5, {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^5] / QPochhammer[ x]^5 - 1) / 5, {x, 0, n}]; (* Michael Somos, Nov 13 2016 *)

x='x+O('x^66); concat([0],Vec(eta(x^5)/eta(x)^5-1)/5) \\ Joerg Arndt, Nov 27 2016

a(n) = A277212(n)/5, n > 0.
G.f.: ((Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^5) - 1)/5.
a(n) ~ exp(4*Pi*sqrt(n/5)) / (sqrt(2) * 5^(11/4) * n^(7/4)). - Vaclav Kotesovec, Nov 10 2016

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

A277283 Expansion of Product_{n>=1} (1 - x^(6*n))/(1 - x^n)^6 in powers of x.

Original entry on oeis.org

1, 6, 27, 98, 315, 918, 2491, 6366, 15498, 36182, 81501, 177876, 377558, 781626, 1582173, 3137832, 6108051, 11687598, 22012816, 40855674, 74799828, 135210868, 241511115, 426570624, 745516240, 1290006276, 2211202692, 3756468658, 6327617862, 10572763842

Offset: 0

Seiichi Manyama, Nov 07 2016

nonn

			G.f.: 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 918*x^5 + 2491*x^6 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
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), A274327 (k=4), A277212 (k=5), this sequence (k=6), A160539 (k=7).

(QPochhammer[x^6, x^6]/QPochhammer[x, x]^6 + O[x]^40)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^(6*k))/(1 - x^k)^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 21 2016 *)

first(n)=my(x='x); Vec(prod(k=1, n, (1-x^(6*k))/(1-x^k)^6, 1+O(x^(n+1)))) \\ Charles R Greathouse IV, Nov 07 2016

G.f.: Product_{n>=1} (1 - x^(6*n))/(1 - x^n)^6.
G.f.: (x^6; x^6)inf/((x; x)_inf)^6, where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2016
a(n) ~ 35*sqrt(35) * exp(sqrt(35*n)*Pi/3) / (3456*sqrt(3)*n^2). - Vaclav Kotesovec, Nov 21 2016

A160459 Omit first term of A160458 and divide by 5.

Original entry on oeis.org

2, 13, 66, 286, 1102, 3879, 12688, 39050, 114114, 318863, 856654, 2222688, 5589916, 13668072, 32576016, 75845402, 172830788, 386088741, 846744800, 1825447086, 3872819904, 8094022001, 16679126516, 33916289400, 68106769602, 135148379654, 265177195950

Offset: 1

N. J. A. Sloane, Nov 13 2009

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Watson, G. N., Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128. See the expression C^2/B^10.

Cf. A160458, A277212, A277974.

x='x+O('x^66); v=Vec((eta(x^5)/eta(x)^5)^2); vector(#v-1,j,v[j+1]/5) \\ Joerg Arndt, Nov 27 2016~

a(n) = 1/5 * A160458(n) = 1/5 * Sum_{k=0..n} A277212(k)*A277212(n-k) = 2 * A277974(n) + 5 * Sum_{k=1..n-1} A277974(k)*A277974(n-k). - Seiichi Manyama, Nov 27 2016

A278668 Expansion of Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^3 in powers of x.

Original entry on oeis.org

1, 3, 9, 22, 51, 107, 218, 420, 788, 1428, 2531, 4375, 7430, 12377, 20313, 32833, 52402, 82585, 128750, 198588, 303428, 459375, 689710, 1027243, 1518709, 2229375, 3251022, 4710777, 6785378, 9717677, 13841991, 19614182, 27656250, 38810312, 54216128, 75406438

Offset: 0

Seiichi Manyama, Nov 25 2016

nonn

			G.f.: 1 + 3*x + 9*x^2 + 22*x^3 + 51*x^4 + 107*x^5 + 218*x^6 + ...

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

Cf. Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^k: A035959 (k=1), A160461 (k=2), this sequence (k=3), A278680 (k=4), A277212 (k=5), A182821 (k=6).

nmax = 30; CoefficientList[Series[Product[(1 - x^(5*k))/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)

G.f.: Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^3.

a(n) ~ exp(2*Pi*sqrt(7*n/15)) * 7^(3/4) / (20 * 3^(3/4) * 5^(1/4) * n^(5/4)). - Vaclav Kotesovec, Nov 10 2017

A278680 Expansion of Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^4 in powers of x.

Original entry on oeis.org

1, 4, 14, 40, 105, 251, 570, 1226, 2540, 5075, 9855, 18630, 34439, 62340, 110805, 193624, 333235, 565415, 947040, 1567130, 2564425, 4152535, 6658711, 10579380, 16663755, 26033200, 40357641, 62106290, 94912385, 144088840, 217368655, 325945320, 485950150, 720515475

Offset: 0

Seiichi Manyama, Nov 25 2016

nonn

			G.f.: 1 + 4*x + 14*x^2 + 40*x^3 + 105*x^4 + 251*x^5 + 570*x^6 + ...

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

Cf. Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^k: A035959 (k=1), A160461 (k=2), A278668 (k=3), this sequence (k=4), A277212 (k=5), A182821 (k=6).

nmax = 30; CoefficientList[Series[Product[(1 - x^(5*k))/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)

G.f.: Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^4.

a(n) ~ 19 * exp(Pi*sqrt(38*n/15)) / (120 * sqrt(10) * n^(3/2)). - Vaclav Kotesovec, Nov 10 2017

cp's OEIS Frontend

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

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A109064 Expansion of eta(q)^5 / eta(q^5) in powers of q.

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A160458 Coefficients in the expansion of C^2/B^10, in Watson's notation of page 106.

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A273845 Expansion of Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^3 in powers of x.

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A277974 Expansion of ((Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^5) - 1)/5 in powers of x.

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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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A277283 Expansion of Product_{n>=1} (1 - x^(6*n))/(1 - x^n)^6 in powers of x.

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