A003105 -id:A003105 - cp's OEIS frontend

A262928 Expansion of Product_{k>=0} (1 + x^(3*k+2)).

1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 2, 1, 1, 2, 1, 1, 3, 1, 2, 3, 1, 3, 4, 2, 4, 4, 2, 5, 5, 3, 7, 5, 4, 8, 6, 6, 10, 7, 7, 12, 8, 10, 14, 9, 12, 16, 11, 16, 19, 13, 19, 21, 16, 24, 25, 19, 28, 28, 23, 35, 32, 28, 40, 36, 34, 48, 42, 41, 55, 47, 49, 65, 55

Offset: 0

Vaclav Kotesovec, Oct 04 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A003105, A035386, A261612.

nmax=100; CoefficientList[Series[Product[(1+x^(3*k-1)),{k,1,nmax}],{x,0,nmax}],x]
nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 3] == 2, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}];], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 13 2017 *)

a(n) ~ exp(sqrt(n)*Pi/3) / (2^(5/3)*sqrt(3)*n^(3/4)) * (1 - (Pi/144 + 9/(8*Pi)) / sqrt(n)). - Vaclav Kotesovec, Oct 04 2015, extended Jan 16 2017

G.f.: Sum_{k>=0} x^(k*(3*k + 1)/2) / Product_{j=1..k} (1 - x^(3*j)). - Ilya Gutkovskiy, Nov 24 2020

A070048 Number of partitions of n into odd parts in which no part appears more than thrice.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 11, 13, 16, 18, 21, 24, 27, 32, 36, 41, 48, 54, 61, 70, 78, 88, 100, 112, 127, 143, 159, 179, 199, 222, 248, 276, 308, 342, 380, 421, 465, 516, 570, 629, 697, 767, 845, 932, 1022, 1124, 1236, 1355, 1488, 1631, 1785, 1954, 2136

Offset: 0

N. J. A. Sloane, May 09 2002

nonn

Also number of partitions of n into distinct parts in which no part is multiple of 4. - Vladeta Jovovic, Jul 31 2004
McKay-Thompson series of class 64a for the Monster group.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + x + x^2 + 2*x^3 + x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + ...
T64a = 1/q + q^7 + q^15 + 2*q^23 + q^31 + 2*q^39 + 3*q^47 + 3*q^55 + 4*q^63 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..220 from Reinhard Zumkeller)
G. E. Andrews and R. P. Lewis, An algebraic identity of F. H. Jackson and its implications for partitions, Discrete Math., 232 (2001), 77-83.
Cristina Ballantine and Mircea Merca, 4-Regular partitions and the pod function, arXiv:2111.10702 [math.CO], 2021.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
M. D. Hirschhorn, J. A. Sellers, A Congruence Modulo 3 for Partitions into Distinct Non-Multiples of Four, Article 14.9.6, Journal of Integer Sequences, Vol. 17 (2014).
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. 12.
Index entries for McKay-Thompson series for Monster simple group

Cf. A042968, A001935, A261734.

Cf. A000700 (m=2), A003105 (m=3), A096938 (m=5), A261770 (m=6), A097793 (m=7), A261771 (m=8), A112193 (m=9), A261772 (m=10).

a070048 = p a042968_list where
   p _      0 = 1
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Oct 01 2012

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^2, x^4], {x, 0, n}]; (* Michael Somos, Jul 01 2014 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^4] / (QPochhammer[ x] QPochhammer[ x^8]), {x, 0, n}]; (* Michael Somos, Jul 01 2014 *)

{a(n) = local(A); if( n<0, 0 ,A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A) / (eta(x + A) * eta(x^8 + A)), n))};

G.f.: Product_{i>0} (1+x^i)/(1+x^(4*i)). - Vladeta Jovovic, Jul 31 2004
Expansion of chi(x) * chi(x^2) = psi(x) / psi(-x^2) = phi(-x^4) / psi(-x) = chi(-x^4) / chi(-x) in powers of x where phi(), psi(), chi() are Ramanujan theta functions. - Michael Somos, Jul 01 2014
Expansion of q^(1/8) * eta(q^2) * eta(q^4) / (eta(q) * eta(q^8)) in powers of q.
Euler transform of period 8 sequence [1, 0, 1, -1, 1, 0, 1, 0, ...].
Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u - v^3) * (u^3 - v) + 3*u*v. - Michael Somos, Jul 01 2014
G.f.: Product_{k>0} (1 - x^(8*k - 4)) / (1 - x^(2*k - 1)).
a(n) ~ exp(sqrt(n)*Pi/2) / (4*n^(3/4)) * (1 - (3/(4*Pi) + Pi/32) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017

Additional comments from Michael Somos, Dec 04 2002

A046042 Number of partitions of n into fourth powers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 9, 9, 9, 9, 9, 9

Offset: 1

Eric W. Weisstein

nonn

In general, the number of partitions of n into perfect s-th powers (s>=1) is asymptotic to (2*Pi)^(-(s+1)/2) * sqrt(s/(s+1)) * k * n^(1/(s+1)-3/2) * exp((s+1)*k*n^(1/(s+1))), where k = (Gamma(1 + 1/s) * Zeta(1 + 1/s) / s)^(s/(s+1)) [Hardy & Ramanujan, 1917]. - Vaclav Kotesovec, Dec 29 2016

			a(33) = 3 because we have [16,16,1], [16,1,1,...,1] (17 1's) and [1,1,...,1] (33 1's).

H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proceedings of the London Mathematical Society, 2, XVI, 1917, p. 373.
Herman P. Robinson, Letter to N. J. A. Sloane, Jan 1974.
Eric Weisstein's World of Mathematics, Partition

Cf. A000583, A002377, A003105.
Cf. A001156, A003108, A046042.
Cf. A037444, A259792, A259793.

a046042 = p $ tail a000583_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, May 18 2015   ~

g:=-1+1/product(1-x^(j^4),j=1..10): gser:=series(g,x=0,105): seq(coeff(gser,x,n),n=1..102); # Emeric Deutsch, Apr 06 2006

g = -1 + 1/Product[1 - x^(j^4), {j, 1, 10}]; gser =
Series[g, {x, 0, 105}]; Table[Coefficient[gser, x, n], {n, 1, 102}] (* Jean-François Alcover, Oct 29 2012, after Emeric Deutsch *)

G.f.: -1+1/product(1-x^(j^4),j=1..infinity). - Emeric Deutsch, Apr 06 2006
a(n) ~ exp(5 * (Gamma(1/4)*Zeta(5/4))^(4/5) * n^(1/5) / 2^(16/5)) * (Gamma(1/4)*Zeta(5/4))^(4/5) / (2^(47/10) * sqrt(5) * Pi^(5/2) * n^(13/10)) [Hardy & Ramanujan, 1917]. - Vaclav Kotesovec, Dec 29 2016
G.f.: Sum_{i>=1} x^(i^4) / Product_{j=1..i} (1 - x^(j^4)). - Ilya Gutkovskiy, May 07 2017

A096938 McKay-Thompson series of class 60F for the Monster group.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 4, 4, 6, 7, 8, 10, 12, 14, 16, 19, 22, 26, 30, 35, 41, 47, 54, 62, 70, 80, 92, 104, 118, 135, 152, 171, 194, 218, 244, 275, 308, 344, 386, 432, 481, 537, 598, 664, 738, 819, 908, 1006, 1114, 1232, 1362, 1503, 1658, 1828, 2012, 2214, 2436, 2676

Offset: 0

Noureddine Chair, Aug 18 2004

nonn

The inverted graded parafermionic partition function.
Also number of partitions of n into parts congruent to {1,3,7,9} mod 10. Also number of partitions of n into odd parts parts in which no part appears more than 4 times.
Number of partitions of n into distinct parts in which no part is a multiple of 5.
This generating function is a generalization of the sequences A003105 and A006950. It arose in my recent work on partial supersymmetry in writing the graded parafermionic partition function in which I obtained a more general formula.

			a(8)=4, the number of partitions into distinct parts that exclude the number 5 because we can write 8=7+1=6+2=4+3+1.
T60F = 1/q + q^5 + q^11 + 2*q^17 + 2*q^23 + 2*q^29 + 3*q^35 + 4*q^41 +...

T. M. Apostol, An Introduction to Analytic Number Theory, Springer-Verlag, NY, 1976

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Cristina Ballantine and Brooke Feigon, Truncated Theta Series Related to the Jacobi Triple Product Identity, arXiv:2401.04019 [math.CO], 2024. See page 16.
Nayandeep Deka Baruah and Abhishek Sarma, Arithmetic properties of 5-regular partitions into distinct parts, arXiv:2411.02978 [math.NT], 2024. See p. 2.
N. Chair, Partition identities from Partial Supersymmetry, arXiv:hep-th/0409011, 2004.
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. 12.
Donald Spector, Duality, partial supersymmetry and arithmetic number theory, arXiv:hep-th/9710002, 1997.
Donald Spector, Duality, partial supersymmetry and arithmetic number theory, J. Math. Phys. Vol. 39, 1998, p. 1919.
Index entries for sequences related to groups
Index entries for McKay-Thompson series for Monster simple group

Cf. A133563.

Cf. A000700 (m=2), A003105 (m=3), A070048 (m=4), A261770 (m=6), A097793 (m=7), A261771 (m=8), A112193 (m=9), A261772 (m=10).

series(product(1/(1-x^k+x^(2*k)-x^(3*k)+x^(4*k)),k+1..150),x=0,100);

CoefficientList[ Series[ Product[1/(1 - x^k + x^(2k) - x^(3k) + x^(4k)), {k, 70}], {x, 0, 60}], x] (* Robert G. Wilson v, Aug 19 2004 *)
nmax = 50; CoefficientList[Series[Product[(1 + x^k) / (1 + x^(5*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)
QP = QPochhammer; s = QP[q^2]*(QP[q^5]/(QP[q]*QP[q^10])) + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 12 2015 *)

{a(n)=local(A); if(n<0,0, A=x*O(x^n); polcoeff( eta(x^2+A)*eta(x^5+A)/eta(x+A)/eta(x^10+A), n))} /* Michael Somos, Jan 18 2005 */

Euler transform of period 10 sequence [1, 0, 1, 0, 0, 0, 1, 0, 1, 0, ...]. - Vladeta Jovovic, Aug 19 2004
Expansion of q^(1/6)eta(q^2)eta(q^5)/(eta(q)eta(q^10)) in powers of q.
Given g.f. A(x), then B(x)=(A(x^6)/x)^2 satisfies 0=f(B(x), B(x^2)) where f(u, v)=(u^3+v^3)(1+uv)-uv(1-uv)^2. - Michael Somos, Jan 18 2005
G.f.: 1/product_{k>=1} (1-x^k+x^(2*k)-x^(3*k)+x^(4*k)) = 1/Product_{k>0} P10(x^k) where P10 is the 10th cyclotomic polynomial.
a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 15^(1/4) * n^(3/4)) * (1 - (3*sqrt(15)/(16*Pi) + Pi/(6*sqrt(15))) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017

Definition corrected by Vladeta Jovovic, Aug 19 2004

More terms from Robert G. Wilson v, Aug 19 2004

A186099 Sum of divisors of n congruent to 1 or 5 mod 6.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 8, 1, 1, 6, 12, 1, 14, 8, 6, 1, 18, 1, 20, 6, 8, 12, 24, 1, 31, 14, 1, 8, 30, 6, 32, 1, 12, 18, 48, 1, 38, 20, 14, 6, 42, 8, 44, 12, 6, 24, 48, 1, 57, 31, 18, 14, 54, 1, 72, 8, 20, 30, 60, 6, 62, 32, 8, 1, 84, 12, 68, 18, 24, 48, 72, 1, 74, 38, 31, 20, 96, 14, 80, 6

Offset: 1

Michael Somos, Feb 12 2011

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f.: x + x^2 + x^3 + x^4 + 6*x^5 + x^6 + 8*x^7 + x^8 + x^9 + 6*x^10 + 12*x^11 +...
L.g.f.: L(x) = x + x^2/2 + x^3/3 + x^4/4 + 6*x^5/5 + x^6/6 + 8*x^7/7 + x^8/8 +...
where exp(L(x)) = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + 3*x^9 +...+ A003105(n)*x^n +...

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000593, A046913, A113957, A116073, A003105, A284098, A284104, A353908.

Cf. A004016, A005882, A005928.

Table[Total[Select[Divisors[n],MemberQ[{1,5},Mod[#,6]]&]],{n,0,100}]  (* Harvey P. Dale, Feb 24 2011 *)
a[ n_] := If[ n < 1, 0, DivisorSum[n, If[ 1 == GCD[#, 6], #, 0] &]]; (* Michael Somos, Jun 27 2017 *)
a[ n_] := If[n < 1, 0, Times @@ (Which[# < 5, 1, True, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger[n])]; (* Michael Somos, Jun 27 2017 *)

{a(n) = sumdiv( n, d, d * (1 == gcd(d, 6) ))};

{a(n) = direuler( p=2, n, 1 / (1 - X) / (1 - (p>3) * p * X)) [n]};

a(n)=sigma(n/2^valuation(n,2)/3^valuation(n,3)) \\ Charles R Greathouse IV, Dec 07 2011

{S(n,x)=sumdiv(n,d,d*(1-x^d)^(n/d))}
{a(n)=n*polcoeff(sum(k=1,n,S(k,x)*x^k/k)+x*O(x^n),n)}
for(n=1,80,print1(a(n),", "))
/* Paul D. Hanna, Feb 17 2013 */

Expansion of (1 + a(x)^2 - 2*a(x^2)^2) / 12 in powers of x where a() is a cubic AGM function.
a(n) is multiplicative with a(2^e) = a(3^e) = 1, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
Equals the logarithmic derivative of A003105, where A003105(n) = number of partitions of n into parts 6*n+1 or 6*n-1. - Paul D. Hanna, Feb 17 2013
L.g.f.: Sum_{n>=1} a(n)*x^n/n  =  Sum_{n>=1} S(n,x)*x^n/n  where S(n,x) = Sum_{d|n} d*(1-x^d)^(n/d). - Paul D. Hanna, Feb 17 2013
a(n) = A284098(n) + A284104(n). - Seiichi Manyama, Mar 24 2017
G.f.: Sum_{n >= 1} x^n*(x^(10*n) + 5*x^(6*n) + 5*x^(4*n) + 1)/(1 - x^(6*n))^2. - Peter Bala, Dec 19 2021
From Amiram Eldar, Dec 30 2022: (Start)
Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-2/2^s)*(1-3/3^s).
Sum_{k=1..n} a(k) ~ c*n^2, where c = Pi^2/36 = 0.274155... (A353908). (End)

A109697 Number of partitions of n into parts each equal to 1 mod 5.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 5, 6, 7, 7, 7, 8, 10, 11, 12, 12, 13, 15, 17, 18, 19, 20, 23, 26, 28, 29, 31, 34, 38, 41, 43, 45, 50, 55, 60, 63, 66, 71, 79, 85, 90, 94, 101, 110, 120, 127, 133, 141, 153, 165, 176, 184, 195, 210, 227, 241, 254, 267, 286, 307, 327

Offset: 0

Erich Friedman, Aug 07 2005

nonn

			a(11)=3 since 11 = 11 = 6+1+1+1+1+1 = 1+1+1+1+1+1+1+1+1+1+1

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000041, A003105, A284097.

Cf. similar sequences of number of partitions of n into parts congruent to 1 mod m: A000009 (m=2), A035382 (m=3), A035451 (m=4), this sequence (m=5), A109701 (m=6), A109703 (m=7), A277090 (m=8).

g:=1/product(1-x^(1+5*j),j=0..25): gser:=series(g,x=0,85): seq(coeff(gser,x,n),n=0..80); # Emeric Deutsch, Mar 30 2006

Table[Count[IntegerPartitions[n],?(Union[Mod[#,5]]=={1}&)],{n,0,75}] (* _Harvey P. Dale, Oct 08 2011 *)

G.f.: 1/product(1-x^(1+5j), j=0..infinity). - Emeric Deutsch, Mar 30 2006
a(n) ~ Gamma(1/5) * exp(Pi*sqrt(2*n/15)) / (2^(8/5) * 3^(1/10) * 5^(2/5) * Pi^(4/5) * n^(3/5)) * (1 - (3*sqrt(3/10)/(5*Pi) + Pi/(120*sqrt(30))) / sqrt(n)). - Vaclav Kotesovec, Feb 27 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A284097(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017
G.f.: Sum_{k>=0} x^k / Product_{j=1..k} (1 - x^(5*j)). - Ilya Gutkovskiy, Jul 17 2019

More terms from Emeric Deutsch, Mar 30 2006

A162891 Expansion of 1 / Product_{k>=1} (1-x^k-x^(2*k)).

Original entry on oeis.org

1, 1, 3, 5, 11, 18, 36, 59, 109, 181, 318, 525, 902, 1481, 2492, 4087, 6788, 11090, 18274, 29776, 48772, 79332, 129411, 210172, 341958, 554728, 900872, 1460298, 2368555, 3837147, 6218652, 10070389, 16311432, 26407350, 42757335, 69208746, 112032256, 181316714

Offset: 0

Franklin T. Adams-Watters, Jul 16 2009

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000045, A000041, A001156, A003105, A263401, A276527.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1/(&*[(1-x^k-x^(2*k)): k in [1..100]]))); // G. C. Greubel, Oct 24 2018

F:= n-> combinat[fibonacci](n+1):
b:= proc(n, i) option remember; `if`(n=0 or i=1, F(n),
      add((t-> b(t, min(t, i-1)))(n-i*j)*F(j), j=0..n/i))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..39);  # Alois P. Heinz, Aug 24 2019

nmax = 50; CoefficientList[Series[1/Product[1-x^k-x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 16 2016 *)

al(n)=Vec(1/prod(k=1,n,1-x^k-x^(2*k)+x*O(x^n)))

a(n) ~ p / (sqrt(5) * r^(n+1)), where r = (sqrt(5)-1)/2 and p = Product_{n>1} 1/(1 - r^n - r^(2*n)) = 4.64451592505133910330213147... . - Vaclav Kotesovec, Nov 16 2016

A097793 McKay-Thompson series of class 56B for the Monster group.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 5, 7, 8, 10, 12, 14, 17, 21, 24, 28, 34, 39, 46, 53, 61, 71, 82, 94, 108, 124, 142, 162, 185, 210, 238, 271, 306, 345, 390, 439, 494, 556, 623, 698, 783, 875, 977, 1092, 1216, 1354, 1508, 1674, 1859, 2064, 2286, 2532, 2803, 3098, 3424

Offset: 0

Michael Somos, Aug 24 2004

nonn

Number of partitions of n into distinct parts not divisible by 7.

Also McKay-Thompson series of class 56C for Monster. - Michel Marcus, Feb 19 2014

			1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 4*x^7 + 5*x^8 + 7*x^9 + 8*x^10 +...
T56B = 1/q + q^3 + q^7 + 2q^11 + 2q^15 + 3q^19 + 4q^23 + 4q^27 +...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
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. 12.
Index entries for McKay-Thompson series for Monster simple group

Cf. A113297.

Cf. A000700 (m=2), A003105 (m=3), A070048 (m=4), A096938 (m=5), A261770 (m=6), A261771 (m=8), A112193 (m=9), A261772 (m=10).

nmax = 50; CoefficientList[Series[Product[(1 + x^k) / (1 + x^(7*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)
QP = QPochhammer; s = QP[q^2]*(QP[q^7]/(QP[q]*QP[q^14])) + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 12 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( prod( k=1, n, 1 + x^k, 1 + A) / prod( k=1, n\7, 1 + x^(7*k), 1 + A), n))}

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^7 + A) / (eta(x + A) * eta(x^14 + A)), n))}

Euler transform of period 14 sequence [ 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, ...].
Expansion of q^(1/4) * eta(q^2) * eta(q^7) / (eta(q) * eta(q^14)) in powers of q.
G.f.: Product_{k>0} (1 + x^k) / (1 + x^(7*k)).
a(n) ~ exp(Pi*sqrt(2*n/7)) / (2 * 14^(1/4) * n^(3/4)) * (1 - (3*sqrt(7)/ (8*Pi*sqrt(2)) + Pi/(4*sqrt(14))) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017

A112193 Coefficients of replicable function number "54b".

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 9, 11, 13, 16, 19, 23, 27, 32, 38, 44, 52, 61, 71, 82, 95, 110, 127, 145, 167, 191, 218, 249, 283, 322, 365, 414, 469, 529, 597, 673, 757, 851, 955, 1071, 1199, 1341, 1499, 1673, 1865, 2078, 2313, 2572, 2857, 3171, 3517, 3897

Offset: 0

Michael Somos, Aug 28 2005

nonn

a(n) is the number of partitions of n into distinct parts where no part is a multiple of 9. - Joerg Arndt, Aug 31 2015

In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^k)/(1 + x^(m*k)), then a(n) ~ exp(Pi*sqrt((m-1)*n/(3*m))) * (m-1)^(1/4) / (2^(3/2) * 3^(1/4) * m^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 31 2015

			G.f. = 1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 6*x^8 + ... _Michael Somos_, Oct 06 2019
G.f. = q^-1 + q^2 + q^5 + 2*q^8 + 2*q^11 + 3*q^14 + 4*q^17 + 5*q^20 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Comm. Algebra 22, No. 13, 5175-5193 (1994).
Index entries for McKay-Thompson series for Monster simple group

Cf. A261733.

Cf. A000700 (m=2), A003105 (m=3), A070048 (m=4), A096938 (m=5), A261770 (m=6), A097793 (m=7), A261771 (m=8), A261772 (m=10).

b:= proc(n, i) option remember;  local r;
      `if`(2*n>i*(i+1)-(j-> 9*j*(j+1))(iquo(i, 9, 'r')), 0,
      `if`(n=0, 1, b(n, i-1)+`if`(i>n or r=0, 0, b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 31 2015

nmax = 50; CoefficientList[Series[Product[(1 + x^k) / (1 + x^(9*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)
b[n_, i_] := b[n, i] = Module[{q, r}, {q, r} = QuotientRemainder[i, 9]; If[2*n > i*(i+1) - 9*q*(q+1), 0, If[n == 0, 1, b[n, i-1] + If[i>n || r == 0, 0, b[n-i, i-1]]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Oct 07 2016, after Alois P. Heinz *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q^2] QPochhammer[ q^9] / (QPochhammer[ q] QPochhammer[ q^18]), {q, 0, n}]; (* Michael Somos, Oct 06 2019 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^9 + A) / (eta(x + A) * eta(x^18 + A)), n))}; /* Michael Somos, Oct 06 2019 */

a(n) ~ exp(2*Pi*sqrt(2*n/3)/3) / (6^(3/4) * n^(3/4)) * (1 - (9*sqrt(3)/ (16*Pi*sqrt(2)) + sqrt(2)*Pi/(9*sqrt(3))) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017
From Michael Somos, Oct 06 2019: (Start)
Expansion of q^(1/3) * eta(q^2) * eta(q^9) / (eta(q) * eta(q^18)) in powers of q.
Euler transform of period 18 sequence [1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...].
G.f. is a period 1 Fourier Series which satisifies f(-1 / (18 t)) = f(t) where q = exp(2 Pi i t).
Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (1 + u*v) * (u^3 + v^3) - u*v * (1 + u^2*v^2).
Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = (w^2 - v) * (u^2 - v) - 2*u*v*w.
Convolution inverse of A261733.
(End)

A261770 Expansion of Product_{k>=1} (1 + x^k) / (1 + x^(6*k)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 8, 9, 11, 13, 16, 19, 22, 26, 30, 35, 41, 47, 55, 63, 73, 84, 96, 110, 125, 143, 162, 184, 208, 235, 266, 300, 338, 380, 427, 479, 536, 600, 670, 748, 834, 929, 1034, 1149, 1277, 1417, 1571, 1740, 1925, 2129, 2351, 2596, 2863

Offset: 0

Vaclav Kotesovec, Aug 31 2015

nonn

a(n) is the number of partitions of n into distinct parts where no part is a multiple of 6. - Joerg Arndt, Aug 31 2015

Alois P. Heinz, Table of n, a(n) for n = 0..10000
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. 12.

Cf. A261736.
Cf. A000700 (m=2), A003105 (m=3), A070048 (m=4), A096938 (m=5), A097793 (m=7), A261771 (m=8), A112193 (m=9), A261772 (m=10).
Column k=6 of A290307.

b:= proc(n, i) option remember;  local r;
      `if`(2*n>i*(i+1)-(j-> 6*j*(j+1))(iquo(i, 6, 'r')), 0,
      `if`(n=0, 1, b(n, i-1)+`if`(i>n or r=0, 0, b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 31 2015

nmax = 50; CoefficientList[Series[Product[(1 + x^k) / (1 + x^(6*k)), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(5*n/2)/3) * 5^(1/4) / (2^(7/4) * sqrt(3) * n^(3/4)) * (1 - (9/(4*Pi*sqrt(10)) + 5*Pi*sqrt(5/2)/144) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017

G.f.: Product_{k>=1} (1 - x^(12*k-6))/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 07 2017

cp's OEIS Frontend

A262928 Expansion of Product_{k>=0} (1 + x^(3*k+2)).

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A070048 Number of partitions of n into odd parts in which no part appears more than thrice.

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A046042 Number of partitions of n into fourth powers.

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A096938 McKay-Thompson series of class 60F for the Monster group.

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A186099 Sum of divisors of n congruent to 1 or 5 mod 6.

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A109697 Number of partitions of n into parts each equal to 1 mod 5.

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