A001935 -id:A001935 - cp's OEIS frontend

A083365 Expansion of psi(x) / phi(x) in powers of x where phi(), psi() are Ramanujan theta functions.

1, -1, 2, -3, 4, -6, 9, -12, 16, -22, 29, -38, 50, -64, 82, -105, 132, -166, 208, -258, 320, -395, 484, -592, 722, -876, 1060, -1280, 1539, -1846, 2210, -2636, 3138, -3728, 4416, -5222, 6163, -7256, 8528, -10006, 11716, -13696, 15986, -18624, 21666, -25169, 29190, -33808, 39104

Offset: 0

Michael Somos, Apr 24 2003

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution square is A079006.
Convolution inverse is A029838.

			G.f. = 1 - x + 2*x^2 - 3*x^3 + 4*x^4 - 6*x^5 + 9*x^6 - 12*x^7 + 16*x^8 - 22*x^9 + ...
G.f. = q - q^9 + 2*q^17 - 3*q^25 + 4*q^33 - 6*q^41 + 9*q^49 - 12*q^57 + 16*q^65 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 221 Entry 1(i).
A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
H. T. Davis, Introduction to nonlinear differential and integral equations, Dover Publications, Inc., New York 1962, p. 170 MR0181773 (31 #6000)

Seiichi Manyama, Table of n, a(n) for n = 0..10000
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162; see Eqs. (9.1),(9.3).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions, q-Pochhammer Symbol

Cf. A001935, A029838, A108494.

(psi(x) / phi(x))^b: this sequence (b=1), A079006 (b=2), A187053 (b=3), A001938 (b=4), A195861 (b=5), A320049 (b=6), A320050 (b=7).

phi[x_] := EllipticTheta[3, 0, x]; psi[x_] := (1/2)*x^(-1/8)*EllipticTheta[2, 0, x^(1/2)]; s = Series[ psi[x]/phi[x], {x, 0, 100}]; A083365 = CoefficientList[s, x] (* Jean-François Alcover, Feb 18 2015 *)
nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k))^2/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 04 2016 *)
(QPochhammer[-x^2, x^2, -1/2] + O[x]^50)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^2] / QPochhammer[ -x, x^2], {x, 0, n}]; (* Michael Somos, Oct 10 2019~ *)

{a(n) = my(A, m); if( n<0, 0, A = 1 + O(x); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A = sqrt(A / (1 + 4 * x * A^2))); polcoeff(sqrt(A), n))};

{a(n) = my(A); if( n<0, 0, A = contfracpnqn( matrix(2, (sqrtint(8*n + 1) + 1)\2, i, j, if( i==1, x^(j-1), 1 + if( j>1, x^(j-1))))); polcoeff(A[2,1] / A[1,1] + x * O(x^n), n))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3, n))};

Expansion of f(-x^4) / f(x) = psi(x) / phi(x) = psi(x^2) / psi(x) = psi(-x) / phi(-x^2) = 1 / (chi(x) * chi(-x^2)) = 1 / (chi^2(x) * chi(-x)) = chi(-x) / chi^2(-x^2) = (psi(x^2) / phi(x))^(1/2) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of k^(1/4) / (2^(1/2) * q^(1/8)) in powers of q where k is elliptic modulus and q is the nome.
Expansion of q^(-1/8) * eta(q) * eta(q^4)^2 / eta(q^2)^3 in powers of q.
Given g.f. A(x), then B(q) = q * A(q^8) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v^2 - u^4 * (1 + 4*v^4).
Given g.f. A(x), then B(q) = q * A(q^8) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = v^4 - u^4 + u*v + 4*(u*v)^3.
Given g.f. A(x), then B(q) = q * A(q^8) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = w - u^2*v*(1 + 2*w^2). - Michael Somos, May 29 2005
Given g.f. A(x), then B(q) = q * A(q^8) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u2*u6 - u1*u3 * (u2^2 + u6^2). - Michael Somos, May 29 2005
Given g.f. A(x), then B(q) = sqrt(2) * q * A(q^8) satisfies 0 = f(B(q), B(q^7)) where f(u, v) = (1 - u^8) * (1 - v^8) - (1 - u*v)^8. - Michael Somos, Jan 01 2006
Euler transform of period 4 sequence [-1, 2, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(-1/2) * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A108494. - Michael Somos, Feb 29 2012
G.f.: Product_{k>0} (1 + x^(2*k)) / (1 + x^(2*k - 1)) = (Sum_{k>0} x^(k^2 - k)) / (Sum_{k>0} x^((k^2 - k)/2)).
G.f.: 1 / (1 + x / (1 + x + x^2 / (1 + x^2 + x^3 / (1 + x^3 + ...)))).
A001935(n) = (-1)^n a(n).
G.f.: (1+1/Q(0))/2, where Q(k)= 1 + x^(k+1) + x^(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 30 2013
a(n) ~ (-1)^n * exp(Pi*sqrt(n/2))/(2^(11/4)*n^(3/4)). - Vaclav Kotesovec, Jul 04 2016
G.f.: (-x^2; x^2){-1/2} = ((-1; x^2){1/2})/2, where (a; q)n is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2016
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A109506(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 14 2017
a(n) ~ (-1)^n * exp(Pi*sqrt(n/2)) / (2^(11/4) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017
G.f.: exp(Sum_{k>=1} (-1)^k*x^k/(k*(1 + x^k))). - Ilya Gutkovskiy, May 28 2018

A104502 Number of partitions where no part is a multiple of 9.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, 54, 74, 96, 128, 165, 216, 275, 354, 447, 569, 712, 896, 1113, 1388, 1712, 2117, 2595, 3186, 3882, 4735, 5739, 6959, 8392, 10121, 12150, 14582, 17429, 20823, 24789, 29494, 34979, 41456, 48993, 57856, 68148, 80204

Offset: 0

Eric W. Weisstein, Mar 11 2005

nonn

Coefficients of the B-Dyson Mod 27 identity.

Also partitions where parts are repeated at most 8 times. - Joerg Arndt, Dec 31 2012

			G.f. = 1 + q + 2*q^2 + 3*q^3 + 5*q^4 + 7*q^5 + 11*q^6 + 15*q^7 + 22*q^8 + 29*q^9 + ...
B(q) = q + q^4 + 2*q^7 + 3*q^10 + 5*q^13 + 7*q^16 + 11*q^19 + 15*q^22 + ...

F. J. Dyson, A walk through Ramanujan's garden, pp. 7-28 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988, see p. 15, eq. (11).

Seiichi Manyama, 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. 15.
Eric Weisstein's World of Mathematics, Dyson Mod 27 Identities

Cf. A062246, A104501, A104503, A104504.
Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.
Cf. A112193, A261733, A320611.

seq(coeff(series(mul((1-x^(9*k))/(1-x^k),k=1..n),x,n+1), x, n), n = 0 .. 50); # Muniru A Asiru, Sep 29 2018

nmax = 50; CoefficientList[Series[Product[(1 - x^(9*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)
a[n_] := a[n] = (1/n) Sum[DivisorSum[k, Boole[!Divisible[#, 9]] #&] a[n-k], {k, 1, n}]; a[0] = 1;
a /@ Range[0, 50] (* Jean-François Alcover, Oct 01 2019, after Seiichi Manyama *)
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 9], 0, 2] ], {n, 0, 46}] (* Robert Price, Jul 29 2020 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^9 + A) / eta(x + A), n))}; /* Michael Somos, Jan 09 2006 */

{A116607(n)=sigma(n)-if(n%9==0, 9*sigma(n/9))}
{a(n)=polcoeff(exp(sum(k=1, n+1, A116607(k)*x^k/k+x*O(x^n))), n)} /* Paul D. Hanna, Jun 13 2011 */

Expansion of q^(-1/3) * eta(q^9) / eta(q) in powers of q. - Michael Somos, Jan 09 2006
Euler transform of period 9 sequence [1, 1, 1, 1, 1, 1, 1, 1, 0, ...]. - Michael Somos, Jan 09 2006
Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^3 + v^3 - u*v - 3*(u*v)^2. - Michael Somos, Jan 09 2006
G.f.: Product_{k>0} (1-x^(9k))/(1-x^k) = 1 + 1/(1-x)*(Sum_{k>0} x^(k^2+k) Product_{i=1..k} (1+x^i+x^(2i))/((1-x^(2i))*(1-x^(2i+1))))
G.f. A(x) = 1/g.f. A062246.
Logarithmic derivative yields A116607 (sum of the divisors of n which are not divisible by 9). - Paul D. Hanna, Jun 13 2011
a(n) ~ 2*Pi * BesselI(1, 4*sqrt(3*n + 1) * Pi/9) / (9*sqrt(3*n + 1)) ~ exp(4*Pi*sqrt(n/3)/3) / (sqrt(2) * 3^(7/4) * n^(3/4)) * (1 + (2*Pi/(9*sqrt(3)) - 9*sqrt(3)/(32*Pi)) / sqrt(n) + (2*Pi^2/243 - 405/(2048*Pi^2) - 5/16) / n). - Vaclav Kotesovec, Aug 31 2015, extended Jan 14 2017
a(n) = (1/n)*Sum_{k=1..n} A116607(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
G.f. is a period 1 Fourier series that satisfies f(-1 / (81 t)) = 1/3 g(t) where g() is the g.f. for A062246. - Michael Somos, Jun 27 2017

Simplified definition. - N. J. A. Sloane, Oct 20 2019

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

A219601 Number of partitions of n in which no parts are multiples of 6.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, 49, 65, 85, 111, 143, 184, 234, 297, 374, 470, 586, 729, 902, 1113, 1367, 1674, 2042, 2485, 3013, 3645, 4395, 5288, 6344, 7595, 9070, 10809, 12852, 15252, 18062, 21352, 25191, 29671, 34884, 40948, 47985, 56146, 65592

Offset: 0

Arkadiusz Wesolowski, Nov 23 2012

Also partitions where parts are repeated at most 5 times. [Joerg Arndt, Dec 31 2012]

			7 = 7
  = 5 + 2
  = 5 + 1 + 1
  = 4 + 3
  = 4 + 2 + 1
  = 4 + 1 + 1 + 1
  = 3 + 3 + 1
  = 3 + 2 + 2
  = 3 + 2 + 1 + 1
  = 3 + 1 + 1 + 1 + 1
  = 2 + 2 + 2 + 1
  = 2 + 2 + 1 + 1 + 1
  = 2 + 1 + 1 + 1 + 1 + 1
  = 1 + 1 + 1 + 1 + 1 + 1 + 1
so a(7) = 14.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Arkadiusz Wesolowski)
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. 15.
Eric Weisstein's World of Mathematics, Partition Function b_k

Cf. A097797.
Cf. A261770, A261736, A320608.
Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

m = 47; f[x_] := (x^6 - 1)/(x - 1); g[x_] := Product[f[x^k], {k, 1, m}]; CoefficientList[Series[g[x], {x, 0, m}], x] (* Arkadiusz Wesolowski, Nov 27 2012 *)
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 6], 0, 2] ], {n, 0, 47}] (* Robert Price, Jul 28 2020 *)

for(n=0, 47, A=x*O(x^n); print1(polcoeff(eta(x^6+A)/eta(x+A), n), ", "))

G.f.: P(x^6)/P(x), where P(x) = prod(k>=1, 1-x^k).
a(n) ~ Pi*sqrt(5) * BesselI(1, sqrt(5*(24*n + 5)/6) * Pi/6) / (3*sqrt(24*n + 5)) ~ exp(Pi*sqrt(5*n)/3) * 5^(1/4) / (12 * n^(3/4)) * (1 + (5^(3/2)*Pi/144 - 9/(8*Pi*sqrt(5))) / sqrt(n) + (125*Pi^2/41472 - 27/(128*Pi^2) - 25/128) / n). - Vaclav Kotesovec, Aug 31 2015, extended Jan 14 2017
a(n) = (1/n)*Sum_{k=1..n} A284326(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

A035985 Number of partitions of n into parts not a multiple of 7. Also number of partitions with at most 6 parts of size 1 and differences between parts at distance 9 are greater than 1.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 14, 21, 28, 39, 51, 70, 90, 119, 153, 199, 252, 324, 406, 515, 642, 804, 994, 1236, 1517, 1869, 2282, 2791, 3387, 4118, 4970, 6006, 7217, 8673, 10374, 12411, 14780, 17601, 20883, 24766, 29274, 34588, 40741, 47964, 56319, 66080, 77350

Offset: 0

Olivier Gérard

Case k=10, i=7 of Gordon Theorem.

			B(x) = x +x^5 +2*x^9 +3*x^13 +5*x^17 +7*x^21 +11*x^25 +14*x^29 +...

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
GDZ, Digitized volumes of Crelle
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. 15.
G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128. This sequence arises as the coefficients of Y = C/B on p. 118.

Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

Cf. A320609.

nmax = 50; CoefficientList[Series[Product[(1 - x^(7*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)
QP = QPochhammer; s = QP[q^7]/QP[q] + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 7], 0, 2] ], {n, 0, 47}] (* Robert Price, Jul 28 2020 *)

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^7+A)/eta(x+A), n))} /* Michael Somos, Jan 17 2006 */

Vec(prod(k=1, 50, (1 - x^(7*k))/(1 - x^k)) + O(x^51)) \\ Indranil Ghosh, Mar 25 2017

A035985_upto(N,q='x+O('x^N))=Vec(eta(q^7)/eta(q)) \\ M. F. Hasler, Dec 09 2019

Euler transform of period 7 sequence [1, 1, 1, 1, 1, 1, 0, ...]. - Michael Somos, Jan 17 2006
Given g.f. A(x), then B(x)=x*A(x^4) satisfies 0=f(B(x), B(x^3)) where f(u, v)=(u^4+v^4)-u*v*(1+3*u*v+7*(u*v)^2).
G.f.: Product_{k>0} (1-x^(7k))/(1-x^k).
Given g.f. A(x) then B(x)=x*A(x)^4 satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u,v,w)= (u^2+u*w+w^2) -v -8*v*(u+v+w) -49*v^2*(u+w). - Michael Somos, May 28 2006
G.f. is product k>0 P7(x^k) where P7 is 7th cyclotomic polynomial.
Expansion of q^(-1/4)eta(q^7)/eta(q) in powers of q. - Michael Somos, Jan 17 2006
a(n) ~ 2*Pi * BesselI(1, sqrt((4*n + 1)/7) * Pi) / (7*sqrt(4*n + 1)) ~ exp(2*Pi*sqrt(n/7)) / (2 * 7^(3/4) * n^(3/4)) * (1 + (Pi/(4*sqrt(7)) - 3*sqrt(7)/(16*Pi)) / sqrt(n) + (Pi^2/224 - 105/(512*Pi^2) - 15/64) / n). - Vaclav Kotesovec, Aug 31 2015, extended Jan 14 2017
a(n) = (1/n)*Sum_{k=1..n} A113957(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

Definition simplified by N. J. A. Sloane, Oct 20 2019

A261776 Expansion of Product_{k>=1} (1 - x^(10*k))/(1 - x^k).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 55, 75, 98, 130, 169, 220, 282, 363, 460, 584, 735, 923, 1151, 1435, 1775, 2194, 2698, 3311, 4045, 4935, 5994, 7270, 8787, 10600, 12749, 15310, 18330, 21912, 26130, 31107, 36949, 43823, 51863, 61290, 72293, 85145, 100107

Offset: 0

Vaclav Kotesovec, Aug 31 2015

nonn

General asymptotic formula (Hagis, 1971): If s > 1 and g.f. = Product_{k>=1} (1 - x^(s*k))/(1 - x^k), then a(n) ~ exp(Pi*sqrt(2*n*(s-1)/(3*s))) * (s-1)^(1/4) / (2 * 6^(1/4) * s^(3/4) * n^(3/4)) * (1 + ((s-1)^(3/2)*Pi/(24*sqrt(6*s)) - 3*sqrt(6*s) / (16*Pi * sqrt(s-1))) / sqrt(n) + ((s-1)^3*Pi^2/(6912*s) - 45*s/(256*(s-1)*Pi^2) - 5*(s-1)/128) / n), minor asymptotic terms added by Vaclav Kotesovec, Jan 13 2017
The formula in the article by Noureddine Chair: "The Euler-Riemann Gases, and Partition Identities", p. 32, is incorrect (must be s -> s-1 and 24 -> 24*n).
Number of partitions in which no part occurs more than 9 times. - Ilya Gutkovskiy, May 31 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Noureddine Chair, The Euler-Riemann Gases, and Partition Identities, arXiv:1306.5415 [math-ph], 2013, p. 32.
Peter Hagis jr., Partitions with a restriction on the multiplicity of the summands, Transactions of the American Mathematical Society, Volume 155, Number 2, April 1971.
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. 15.

Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

Cf. A261772, A145707, A320612.

nmax = 50; CoefficientList[Series[Product[(1 - x^(10*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 10], 0, 2] ], {n, 0, 47}] (* Robert Price, Jul 29 2020 *)

Vec(prod(k=1, 51, (1 - x^(10*k))/(1 - x^k)) + O(x^51)) \\ Indranil Ghosh, Mar 25 2017

a(n) ~ 3*Pi * BesselI(1, sqrt((24*n + 9)/10) * Pi/2) / (5*sqrt(24*n + 9)) ~ exp(Pi*sqrt(3*n/5)) * 3^(1/4) / (4 * 5^(3/4) * n^(3/4)) * (1 + (3^(3/2)*Pi/(16*sqrt(5)) - sqrt(15)/(8*Pi)) / sqrt(n) + (27*Pi^2/2560 - 25/(128*Pi^2) - 45/128) / n). - Vaclav Kotesovec, Aug 31 2015, extended Jan 14 2017

a(n) = (1/n)*Sum_{k=1..n} A284344(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

A174713 Triangle read by rows, A173305 (A000009 shifted down twice) * A174712 (diagonalized variant of A000041).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 2, 4, 2, 2, 3, 5, 3, 4, 3, 6, 4, 4, 3, 5, 8, 5, 6, 6, 5, 10, 6, 8, 6, 5, 7, 12, 8, 10, 9, 10, 7, 15, 10, 12, 12, 10, 7, 11, 18, 12, 16, 15, 15, 14, 11, 22, 15, 20, 18, 20, 14, 11, 15

Offset: 0

Gary W. Adamson, Mar 27 2010

Row sums = A000041, the partition numbers.
The current triangle is the 2nd in an infinite set, followed by A174714 (k=3), and A174715, (k=4); in which row sums of each triangle = A000041.
k-th triangle in the infinite set can be defined as having the sequence:
"Euler transform of ones: (1,1,1,...) interleaved with (k-1) zeros"; shifted down k times (except column 0) in successive columns, then multiplied * triangle A174712, the diagonalized variant of A000041, A174713 begins with A000009 shifted down twice (triangle A173305); where A000009 = the Euler transform of period 2 sequence: [1,0,1,0,...].
Similarly, triangle A174714 begins with A000716 shifted down thrice; where A000716 = the Euler transform of period 3 series: [1,1,0,1,1,0,...]. Then multiply the latter as an infinite lower triangular matrix * A174712, the diagonalized variant of A000041, obtaining triangle A174714 with row sums = A000041.
Case k=4 = triangle A174715 which begins with the Euler transform of period 4 series: [1,1,1,0,1,1,1,0,...], shifted down 4 times in successive columns then multiplied * A174712, the diagonalized variant of A000041.
All triangles in the infinite set have row sums = A000041.
The sequences: "Euler transform of ones interleaved with (k-1) zeros" have the following properties, beginning with k=2:
...
k=2, A000009: = Euler transform of [1,0,1,0,1,0,...] and satisfies
.....A000009. = p(x)/p(x^2), where p(x) = polcoeff A000041; and A000041 =
.....A000009(x) = r(x), then p(x) = r(x) * r(x^2) * r(x^4) * r(x^8) * ...
...
k=3, A000726: = Euler transform of [1,1,0,1,1,0,...] and satisfies
.....A000726(x): = p(x)/p(x^3), and given s(x) = polcoeff A000726, we get
.....A000041(x) = p(x) = s(x) * s(x^3) * s(x^9) * s(x^27) * ...
...
k=4, A001935: = Euler transform of [1,1,1,0,1,1,1,0,...] and satisfies
.....A001935(x) = p(x)/p(x^4) and given t(x) = polcoeff A001935, we get
.....A000041(x) = p(x) = t(x) * t(x^4) * t(x^16) * t(x^64) * ...
...
Also the number of integer partitions of n whose even parts sum to k, for k an even number from zero to n. The version including odd k is A113686. - Gus Wiseman, Oct 23 2023

			First few rows of the triangle =
1;
1;
1, 1;
2, 1;
2, 1, 2;
3, 2, 2;
4, 2, 2, 3;
5, 3, 4, 3;
6, 4, 4, 3, 5;
8, 5, 6, 6, 5;
10, 6, 8, 6, 5, 7;
12, 8, 10, 9, 10, 7;
15, 10, 12, 12, 10, 7, 11;
18, 12, 16, 15, 15, 14, 11;
22, 15, 20, 18, 20, 14, 11, 15;
...
From _Gus Wiseman_, Oct 23 2023: (Start)
Row n = 9 counts the following partitions:
  (9)          (72)        (54)       (63)      (81)
  (711)        (5211)      (522)      (6111)    (621)
  (531)        (3321)      (4311)     (432)     (441)
  (51111)      (321111)    (411111)   (42111)   (4221)
  (333)        (21111111)  (32211)    (3222)    (22221)
  (33111)                  (2211111)  (222111)
  (3111111)
  (111111111)
(End)

Cf. A000009, A173305, A174712, A174714, A174715.
Row sums are A000041.
The odd version is A365067.
The corresponding rank statistic is A366531, odd version A366528.
A000009 counts partitions into odd parts, ranks A066208.
A113685 counts partitions by sum of odd parts, even version A113686.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
Cf. A035363, A066967, A130780, A171966, A182616, A366527, A366533.

Table[Length[Select[IntegerPartitions[n],Total[Select[#,EvenQ]]==k&]],{n,0,15},{k,0,n,2}] (* Gus Wiseman, Oct 23 2023 *)

As infinite lower triangular matrices, A173305 * A174712.

T(n,k) = A000009(n-2k) * A000041(k). - Gus Wiseman, Oct 23 2023

A261775 Expansion of Product_{k>=1} (1 - x^(8*k))/(1 - x^k).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, 53, 72, 94, 124, 161, 208, 266, 341, 431, 545, 684, 856, 1064, 1322, 1631, 2009, 2464, 3014, 3672, 4467, 5411, 6543, 7888, 9489, 11383, 13632, 16280, 19409, 23088, 27415, 32483, 38430, 45371, 53485, 62939, 73950, 86742

Offset: 0

Vaclav Kotesovec, Aug 31 2015

nonn

Number of partitions in which no part occurs more than 7 times. - Ilya Gutkovskiy, May 31 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 30
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. 15.

Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

Cf. A261771, A261735, A320610.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(d*
     signum(irem(d, 8)), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 07 2022

nmax = 50; CoefficientList[Series[Product[(1 - x^(8*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 8], 0, 2] ], {n, 0, 47}] (* Robert Price, Jul 28 2020 *)

Vec(prod(k=1, 51, (1 - x^(8*k))/(1 - x^k)) + O(x^51)) \\ Indranil Ghosh, Mar 25 2017

a(n) ~ Pi*sqrt(7) * BesselI(1, sqrt(7*(24*n + 7)/8) * Pi/6) / (4*sqrt(24*n + 7)) ~ exp(Pi*sqrt(7*n/3)/2) * 7^(1/4) / (2^(7/2) * 3^(1/4) * n^(3/4)) * (1 + (7^(3/2)*Pi/(96*sqrt(3)) - 3*sqrt(3)/(4*Pi*sqrt(7))) / sqrt(n) + (343*Pi^2/55296 - 45/(224*Pi^2) - 35/128) / n). - Vaclav Kotesovec, Aug 31 2015, extended Jan 14 2017
a(n) = (1/n)*Sum_{k=1..n} A284341(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
G.f.: A(x)*A(x^2)*A(x^4) where A(x) is the o.g.f. for A000009. (see Flajolet, Sedgewick link) - Geoffrey Critzer, Aug 07 2022

A232464 Number of compositions of n avoiding the pattern 1111.

Original entry on oeis.org

1, 1, 2, 4, 7, 15, 26, 52, 93, 173, 310, 556, 1041, 1789, 3098, 5620, 9725, 16377, 28764, 48518, 82889, 137161, 237502, 390084, 646347, 1055975, 1774036, 2907822, 4698733, 7581093, 12381660, 19891026, 32113631, 51110319, 80777888, 130175410, 204813395

Offset: 0

Alois P. Heinz, Nov 24 2013

nonn

Number of compositions of n into parts with multiplicity <= 3.

			a(5) = 15: [5], [4,1], [3,2], [2,3], [1,4], [1,2,2], [2,1,2], [1,1,3], [3,1,1], [2,2,1], [1,3,1], [1,2,1,1], [2,1,1,1], [1,1,2,1], [1,1,1,2].
a(6) = 26: [6], [3,3], [5,1], [4,2], [2,4], [1,5], [4,1,1], [3,2,1], [2,3,1], [1,4,1], [3,1,2], [2,2,2], [1,3,2], [1,2,3], [2,1,3], [1,1,4], [1,2,2,1], [2,1,2,1], [1,1,3,1], [3,1,1,1], [2,2,1,1], [1,3,1,1], [1,2,1,2], [2,1,1,2], [1,1,2,2], [1,1,1,3].

Alois P. Heinz, Table of n, a(n) for n = 0..3000

Cf. A001935 (partitions avoiding 1111), A032020 (pattern 11), A232432 (pattern 111), A232394 (consecutive pattern 1111).

Column k=3 of A243081.

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
      add(b(n-i*j, i-1, p+j)/j!, j=0..min(n/i, 3))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);

f[list_]:=Apply[And,Table[Count[list,i]<4,{i,1,Max[list]}]];
g[list_]:=Length[list]!/Apply[Times,Table[Count[list,i]!,{i,1,Max[list]}]];
a[n_] := If[n == 0, 1, Total[Map[g, Select[IntegerPartitions[n], f]]]];
Table[a[n], {n, 0, 40}] (* Geoffrey Critzer, Nov 25 2013, updated by Jean-François Alcover, Nov 20 2023 *)

A328545 Number of 11-regular partitions of n (no part is a multiple of 11).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 55, 76, 99, 132, 171, 224, 286, 370, 468, 597, 750, 945, 1177, 1472, 1820, 2255, 2772, 3410, 4165, 5092, 6185, 7515, 9085, 10978, 13207, 15884, 19025, 22774, 27170, 32388, 38489, 45705, 54120, 64030, 75569, 89100

Offset: 0

N. J. A. Sloane, Oct 19 2019

nonn

Kathiravan, T., and S. N. Fathima. "On L-regular bipartitions modulo L." The Ramanujan Journal 44.3 (2017): 549-558.

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

Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

f:=(k,M) -> mul(1-q^(k*j),j=1..M);
LRP := (L,M) -> f(L,M)/f(1,M);
s := L -> seriestolist(series(LRP(L,80),q,60));
s(11);

Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 11], 0, 2] ], {n, 0, 46}]

a(n) ~ exp(Pi*sqrt(2*n*(s-1)/(3*s))) * (s-1)^(1/4) / (2 * 6^(1/4) * s^(3/4) * n^(3/4)) * (1 + ((s-1)^(3/2)*Pi/(24*sqrt(6*s)) - 3*sqrt(6*s) / (16*Pi * sqrt(s-1))) / sqrt(n) + ((s-1)^3*Pi^2/(6912*s) - 45*s/(256*(s-1)*Pi^2) - 5*(s-1)/128) / n), set s=11. - Vaclav Kotesovec, Aug 01 2022

cp's OEIS Frontend

A083365 Expansion of psi(x) / phi(x) in powers of x where phi(), psi() are Ramanujan theta functions.

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A104502 Number of partitions where no part is a multiple of 9.

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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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A219601 Number of partitions of n in which no parts are multiples of 6.

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A035985 Number of partitions of n into parts not a multiple of 7. Also number of partitions with at most 6 parts of size 1 and differences between parts at distance 9 are greater than 1.

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A261776 Expansion of Product_{k>=1} (1 - x^(10*k))/(1 - x^k).

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