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A104274 Number of partitions of n in which odd squares occur with 2 types c,c* and with multiplicity 1. The even squares and parts that are twice the squares they occur with multiplicity 1.

1, 2, 2, 2, 2, 2, 2, 2, 2, 4, 6, 6, 6, 6, 6, 6, 6, 6, 8, 10, 10, 10, 10, 10, 10, 12, 14, 16, 18, 18, 18, 18, 18, 18, 22, 26, 28, 30, 30, 30, 30, 30, 30, 34, 38, 40, 42, 42, 42, 44, 48, 50, 54, 58, 60, 62, 62, 62, 66, 74, 78, 82, 86, 88, 90, 90, 90

Offset: 0

Noureddine Chair, Feb 27 2005

Convolution of A167700 and A167661. - Vaclav Kotesovec, Sep 19 2017

			E.g. a(10)=6 because we can write it as 91,91*,9*1,9*1*,82,811*.

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

Cf. A080054, A292563.

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

nmax = 100; CoefficientList[Series[Product[(1 + x^((2*k-1)^2)) / (1 - x^((2*k-1)^2)), {k, 1, Floor[Sqrt[nmax]/2] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 19 2017 *)

G.f.: product_{k>0}((1+x^(2k-1)^2)/(1-x^(2k-1)^2)).

a(n) ~ exp(3 * 2^(-8/3) * Pi^(1/3) * ((4-sqrt(2)) * Zeta(3/2))^(2/3) * n^(1/3)) * ((4-sqrt(2)) * Zeta(3/2))^(1/3) / (2^(7/3) * sqrt(3) * Pi^(1/3) * n^(5/6)). - Vaclav Kotesovec, Sep 19 2017

A104276 Number of partitions of n in which both even and odd squares occur with multiplicity 1. There is no restriction on the parts which are twice even squares.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 2, 3, 1, 0, 2, 3, 1, 0, 2, 5, 3, 0, 2, 5, 3, 0, 3, 6, 4, 1, 4, 7, 4, 1, 4, 9, 6, 1, 4, 10, 7, 1, 5, 12, 9, 2, 6, 13, 9, 2, 6, 15, 12, 3, 6, 17, 14, 3, 8, 20, 16, 4, 9, 21, 17, 5, 10, 25, 22, 7, 10, 27, 24, 7, 12, 32, 28, 9, 14, 34, 30, 10, 15, 39, 37

Offset: 0

Noureddine Chair, Mar 01 2005

nonn

			E.g. a(30) = 3 because we can write 30 as 25+4+1 = 16+9+4+1 = 8+8+9+4+1.

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

G.f.: Product_{k>0} ((1+x^(2k-1)^2)/(1-x^(2k)^2) = Product_{k>0} ((1+x^(2k-1)^2)*(1+x^(2k)^2)))/(1-x^2(2k)^2).

A104277 Number of partitions of n in which both even and odd squares occur with multiplicity 1. There is no restriction on the parts which are twice odd squares.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 8, 10, 10, 11, 11, 13, 13, 14, 14, 14, 16, 16, 18, 18, 20, 20, 22, 23, 23, 25, 25, 28, 28, 30, 30, 33, 35, 35, 38, 39, 43, 43, 46, 46, 49, 51, 51, 55, 56, 60, 61

Offset: 0

Noureddine Chair, Mar 01 2005

			a(21)=7 because we can write 21 as 18+2+1 = 16+4+1 = 16+2+2+1 = 9+4+2+2+2+2 = 9+2+2+2+2+2+2 = 4+2+2+2+2+2+2+2+2+1 = 2+2+2+2+2+2+2+2+2+2+1.

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

G.f.: Product_{k>0} (1+x^((2*k)^2))/(1-x^((2*k-1)^2)).

Missing term a(46) added by Jason Yuen, Jan 20 2025

A103265 Number of partitions of n in which both even and odd square parts occur in 2 forms c, c* and with multiplicity 1. There is no restriction on parts which are twice squares.

Original entry on oeis.org

1, 2, 2, 2, 4, 6, 6, 6, 8, 12, 14, 14, 16, 22, 26, 26, 30, 38, 44, 46, 52, 62, 70, 74, 80, 96, 110, 116, 124, 146, 166, 174, 186, 210, 238, 254, 272, 302, 338, 362, 384, 426, 470, 502, 532, 588, 646, 686, 726, 792, 872, 926, 980, 1062

Offset: 0

Noureddine Chair, Feb 27 2005

Convolution of A001156 and A033461. - Vaclav Kotesovec, Aug 18 2015

			E.g. a(8)=8 because 8 can be written as 8, 44*, 422, 4*22, 4211*, 4*211*, 2222, 22211*.

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

Cf. A001156, A015128, A033461, A280263, A279227, A306147.

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

nmax = 50; CoefficientList[Series[Product[(1+x^(k^2)) / (1-x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 18 2015 *)

G.f.: Product_{k>0}((1+x^k^2)/(1-x^k^2)).

a(n) ~ exp(3 * ((4-sqrt(2))*zeta(3/2))^(2/3) * Pi^(1/3) * n^(1/3) / 4) * ((4-sqrt(2))*zeta(3/2))^(2/3) / (2^(7/2) * sqrt(3) * Pi^(7/6) * n^(7/6)). - Vaclav Kotesovec, Dec 29 2016

A102346 Number of partitions of 2n in which odd parts and multiples of 3 and 5 occur with even multiplicities. There is no restriction on the other even parts.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 46, 69, 101, 146, 208, 293, 408, 563, 769, 1042, 1401, 1871, 2482, 3273, 4291, 5596, 7261, 9378, 12057, 15437, 19684, 25005, 31648, 39919, 50184, 62890, 78573, 97883, 121597, 150653, 186169, 229487, 282204, 346230, 423831, 517706

Offset: 0

Noureddine Chair, Feb 21 2005

nonn

			a(5) = 19: [8,2], [8,1,1], [5,5], [4,4,2], [4,4,1,1], [4,2,2,2], [4,2,2,1,1], [4,2,1,1,1,1], [4,3,3], [3,3,2,2], [3,3,2,1,1], [3,3,1,1,1,1], [4,1,1,1,1,1,1], [2,2,2,2,2], [2,2,2,2,1,1], [2,2,2,1,1,1,1], [2,2,1,1,1,1,1,1], [2,1,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1,1].

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.

Cf. A098151.

nmax = 50; CoefficientList[Series[Product[(1+x^k)*(1+x^(15*k))/((1-x^k)*(1+x^(3*k))*(1+x^(5*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 01 2015 *)

q='q+O('q^33); E(k)=eta(q^k);
Vec( (E(2)*E(3)*E(5)*E(30)) / (E(1)^2*E(6)*E(10)*E(15)) ) \\ Joerg Arndt, Sep 01 2015

G.f.: Product((1+x^k)*(1+x^(15*k))/((1-x^k)*(1+x^(3*k))*(1+x^(5*k))), k>=1).
a(n) ~ exp(Pi*sqrt(38*n/5)/3) * sqrt(19) / (12*sqrt(5)*n). - Vaclav Kotesovec, Sep 01 2015
G.f.: (E(2)*E(3)*E(5)*E(30)) / (E(1)^2*E(6)*E(10)*E(15)) where E(k) = prod(n>=1, 1-q^k ). - Joerg Arndt, Sep 01 2015

Corrected by Vladeta Jovovic, Feb 21 2005

Offset and example corrected by Vaclav Kotesovec, Sep 01 2015

A103263 Number of partitions of n into distinct parts prime to 3 and 5.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 5, 5, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 17, 18, 20, 22, 25, 28, 30, 33, 36, 39, 43, 48, 52, 56, 61, 67, 73, 80, 87, 94, 101, 110, 120, 130, 141, 152, 164, 177, 192, 207, 223, 240, 258, 278, 301, 324, 348, 373, 400, 429, 461, 496

Offset: 0

Noureddine Chair, Feb 21 2005

			E.g. a(15)=5 because we can write 15 as 14+1=13+2=11+4=8+7=8+4+2+1.

series(product((1+x^k)*(1+x^(15*k))/((1+x^(3*k))*(1+x^(5*k))),k=1..100),x=0,100);

CoefficientList[ Series[ Product[(1 + x^k)(1 + x^(15*k))/((1 + x^(3k))*(1 + x^(5k))), {k, 100}], {x, 0, 75}], x] (* Robert G. Wilson v, Feb 22 2005 *)

{a(n)=local(A); if (n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)*eta(x^3+A)*eta(x^5+A)*eta(x^30+A)/ (eta(x+A)*eta(x^6+A)*eta(x^10+A)*eta(x^15+A)), n))} /* Michael Somos, Sep 22 2005 */

Expansion of q^(-1/3)(eta(q^2)*eta(q^3)*eta(q^5)*eta(q^30))/(eta(q)*eta(q^6)*eta(q^10)*eta(q^15)) in powers of q. - Michael Somos, Sep 22 2005.
G.f.: product_{k>0}((1+x^k)*(1+x^(15k)))/((1+x^(3k))*(1+x^(5k))).
Euler transform of period 30 sequence [1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, ...]. - Michael Somos, Sep 22 2005
Given g.f. A(x), then B(x)=x*A(x^3) satisfies 0=f(B(x), B(x^2)) where f(u, v)=u*(u-v^2)^2 +v*(v-u^2)^2 -u*v -(u*v)^3. - Michael Somos, Sep 22 2005
Given g.f. A(x), then B(x)=x*A(x^3) satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=(v+u*w)^2 -v*(u^2+w^2). - Michael Somos, Sep 22 2005
G.f.: Product_{k>0} (1+x^k-x^(3k)-x^(4k)-x^(5k)+x^(7k)+x^(8k)). - Michael Somos Sep 22 2005
a(n) ~ exp(2*Pi*sqrt(2*n/5)/3) / (2^(3/4) * sqrt(3) * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 06 2015

More terms from Robert G. Wilson v, Feb 22 2005

A103264 Number of partitions of n into distinct parts prime to 3, 5 and 7.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 5, 6, 7, 8, 8, 9, 9, 10, 11, 13, 14, 15, 16, 18, 19, 21, 23, 24, 26, 28, 31, 34, 37, 39, 42, 45, 49, 53, 56, 60, 64, 69, 75, 81, 86, 92, 98, 105, 113, 122, 130, 138, 147, 157, 168, 179, 191, 202, 215, 230, 246, 262, 279

Offset: 0

Noureddine Chair, Feb 21 2005

nonn

			a(19)=5 because 19 = 17 + 2 = 16 + 2 + 1 = 13 + 4 + 2 = 11 + 8.

series(product((1+x^k)*(1+x^(15*k))*(1+x^(21*k))*(1+x^(35*k)))/((1+x^(3*k))*(1+x^(5*k))*(1+x^(7*k))*(1+x^(105*k))),k=1..100),x=0,100);

CoefficientList[ Series[ Product[(1 + x^k)(1 + x^(15k))(1 + x^(21k))(1 + x^(35k))/((1 + x^(3k))(1 + x^(5k))(1 + x^(7k))(1 + x^(105k))), {k, 100}], {x, 0, 73}], x] (* Robert G. Wilson v, Feb 22 2005 *)

G.f.: product_{k>0}((1+x^k)*(1+x^(15k))*(1+x^(21k))*(1+x^(35k)))/((1+x^(3k))*(1+x^(5k))*(1+x^(7k))*(1+x^(105k))).

a(n) ~ exp(4*Pi*sqrt(n/105)) / (sqrt(2) * 105^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015

More terms from Robert G. Wilson v, Feb 22 2005

A103262 McKay-Thompson series of class 36g for the Monster group.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 11, 16, 21, 26, 34, 44, 58, 74, 93, 116, 143, 178, 221, 272, 332, 402, 487, 588, 710, 854, 1021, 1216, 1444, 1714, 2031, 2400, 2826, 3318, 3888, 4552, 5322, 6208, 7224, 8388, 9726, 11264, 13028, 15044, 17339, 19952, 22930, 26324, 30186

Offset: 0

Noureddine Chair, Feb 21 2005

nonn

Number of partitions of n into distinct parts prime to 3, with 2 types of each part.

This is also the number of partitions of n into parts with 2 types congruent to 1 or 5 mod(6).

			E.g., a(5)=8 because we have 5,5*,41,41*,4*1,4*1*,22*1,22*1* with all parts prime to 3. The parts congruent to 1,5 mod(6) are 5, 5*, 11111, 11111*, 1111*1*, 111*1*1*, 11*1*1*1*, 1*1*1*1*1*.
T36g = 1/q + 2*q^5 + 3*q^11 + 4*q^17 + 5*q^23 + 8*q^29 + 11*q^35 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
Index entries for McKay-Thompson series for Monster simple group

Cf. A003105.

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

CoefficientList[ Series[ Product[(1 + x^k)^2/(1 + x^(3k))^2, {k, 60}], {x, 0, 50}], x] (* Robert G. Wilson v, Feb 22 2005 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(1/6)(eta[q^2]eta[q^3]/(eta[q]eta[q^6]))^2, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 06 2018 *)

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)*eta(x^3+A)/eta(x+A)/eta(x^6+A))^2, n))} /* Michael Somos, Sep 10 2005 */

G.f.: product_{k>0}((1+x^k)/(1+x^(3k)))^2= 1/product_{k>0}((1-x^(6k-1))*(1-x^(6k-5)))^2.
Expansion of q^(1/6)(eta(q^2)eta(q^3)/(eta(q)eta(q^6)))^2 in powers of q.
Euler transform of period 6 sequence [2, 0, 0, 0, 2, 0, ...]. - Michael Somos, Sep 10 2005
a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Sep 01 2015

More terms from Robert G. Wilson v, Feb 22 2005

A100989 Number of partitions of n into parts free of odd hexagonal numbers and the only number with multiplicity in the unrestricted partitions is the number 2 with multiplicity of the form 3k+l, where k is a positive integer and l=0,1.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 3, 4, 6, 6, 9, 11, 13, 16, 20, 20, 23, 29, 35, 41, 49, 59, 68, 82, 96, 112, 131, 154, 178, 207, 242, 277, 321, 371, 425, 489, 562, 641, 733, 839, 953, 1086, 1236, 1399, 1588, 1798, 2032, 2295, 2592, 2917, 3285

Offset: 1

Noureddine Chair, Nov 29 2004

nonn

			a(15)=20 because 15 =13+2 =12+3 =11+4 =10+5 =10+3+2 =9+6=9+4+2 =8+7 =8+5+2 =8+4+3 =7+6+2 =7+5+3 =6+5+4 =6+4+3+2 =9+2+2+2 =7+2+2+2+2 =6+3+2+2+2 =5+4+2+2+2 =4+3+2+2+2+2 =3+2+2+2+2+2+2"

Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.

Cf. A100926, A100927.

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

G.f.: product_{k>0}(1+x^k)/(1-(-1)^k*x^(2*k^2-k)).

A103261 Number of partitions of 2n into parts with 10 types c^1 c^2...C^10 of each part. The even parts appear with multiplicity 1 for each type . The odd parts occur with multiplicity 2 for each part.

Original entry on oeis.org

1, 20, 200, 1360, 7200, 32024, 125280, 443680, 1450240, 4435940, 12827888, 35346800, 93377920, 237675640, 585229760, 1398704736, 3253934080, 7386124520, 16392493800, 35634450320, 75992326592, 159199081600, 328027789600

Offset: 0

Noureddine Chair, Feb 16 2005

nonn

This is also Sequence(A080054)^(10) or sequence(A007096)^(5).

In general, if j > 0 and g.f. = Product_{k>=0} ((1 + x^(2*k+1))/(1 - x^(2*k+1)))^j, then a(n) ~ exp(Pi*sqrt(j*n/2)) * j^(1/4) / (2^(j/2 + 7/4) * n^(3/4)). - Vaclav Kotesovec, Aug 28 2015

			a(2)=200 because we have 10 types of 4, 45 ways of writing 4 in terms of ten of 2's only or ten of 11's only and 100 ways of writing 2's combined with 11's so the total number of ways of writing 4 is 200.

Cf. A080054 (j=1), A007096 (j=2), A261647 (j=3), A014969 (j=4), A261648 (j=5), A014970 (j=6), A014972 (j=8).

series(product(((1+x^k)*(1-x^(2*k)))^(10)/((1-x^k)*(1+x^(2*k)))^(10),k=1..100),x=0,100);

nmax=60; CoefficientList[Series[Product[((1+x^(2*k+1))/(1-x^(2*k+1)))^10,{k,0,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Aug 28 2015 *)

G.f.:(theta_4(0, x^2)/theta_4(0, x))^10= (theta_3(0, x)/theta_4(0, x))^5.

a(n) ~ exp(Pi*sqrt(5*n)) * 5^(1/4) / (64 * sqrt(2) * n^(3/4)). - Vaclav Kotesovec, Aug 28 2015

Example corrected by Vaclav Kotesovec, Sep 01 2015

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User: Noureddine Chair

A104274 Number of partitions of n in which odd squares occur with 2 types c,c* and with multiplicity 1. The even squares and parts that are twice the squares they occur with multiplicity 1.

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A104276 Number of partitions of n in which both even and odd squares occur with multiplicity 1. There is no restriction on the parts which are twice even squares.

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A104277 Number of partitions of n in which both even and odd squares occur with multiplicity 1. There is no restriction on the parts which are twice odd squares.

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A103265 Number of partitions of n in which both even and odd square parts occur in 2 forms c, c* and with multiplicity 1. There is no restriction on parts which are twice squares.

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A102346 Number of partitions of 2n in which odd parts and multiples of 3 and 5 occur with even multiplicities. There is no restriction on the other even parts.

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A103263 Number of partitions of n into distinct parts prime to 3 and 5.

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A103264 Number of partitions of n into distinct parts prime to 3, 5 and 7.

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A103262 McKay-Thompson series of class 36g for the Monster group.

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