A002513 -id:A002513 - cp's OEIS frontend

A029863 Expansion of Product_{k >= 1} 1/(1-x^k)^c(k), where c(1), c(2), ... = 2 3 2 3 2 3 2 3 ....

1, 2, 6, 12, 27, 50, 98, 172, 310, 522, 888, 1444, 2357, 3724, 5882, 9072, 13957, 21082, 31732, 47072, 69545, 101540, 147620, 212516, 304631, 433054, 613030, 861616, 1206089, 1677766, 2324844, 3203748, 4398602, 6009390, 8181250

Offset: 0

N. J. A. Sloane

nonn

Number of partitions of n where there are 2 kinds of odd parts and 3 kinds of even parts. - Ilya Gutkovskiy, Jan 17 2018

			G.f. = 1 + 2*x + 6*x^2 + 12*x^3 + 27*x^4 + 50*x^5 + 98*x^6 + 172*x^7 + ...

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016, p. 16.
Songxin Liang & Jingzhong Zhang, A complete discrimination system for polynomials with complex coefficients and its automatic generation, 1999, p. 2. the sequence Cn is the numbers of cases for root classification of an n-degree polynomial with complex coefficients.
N. J. A. Sloane, Transforms

Cf. A002513, A010693, A085140, A262380.

nmax = 50; CoefficientList[Series[Product[1/((1 + x^k)*(1 - x^k)^3), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 20 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( 1 / (eta(x + A)^2 * eta(x^2 + A)), n))};

Euler transform of period 2 sequence [2, 3, ...].
a(n) ~ 5 * exp(sqrt(5*n/3)*Pi) / (48 * n^(3/2)). - Vaclav Kotesovec, Sep 20 2015
G.f.: Product_{k >= 1} 1/(1-x^k)^A010693(k-1). - Georg Fischer, Dec 10 2020

A262380 Expansion of Product_{k>=1} 1/((1+x^k)*(1-x^k)^4).

Original entry on oeis.org

1, 3, 10, 25, 62, 136, 293, 590, 1165, 2205, 4097, 7391, 13120, 22780, 38997, 65613, 109036, 178660, 289575, 463842, 735870, 1155717, 1799620, 2777795, 4254859, 6467115, 9761770, 14633605, 21799465, 32273399, 47506759, 69537814, 101252595, 146675875, 211451893

Offset: 0

Vaclav Kotesovec, Sep 20 2015

nonn

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

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. 16.

Cf. A002513 (m=2), A029863 (m=3), A261998.

nmax = 50; CoefficientList[Series[Product[1/((1 + x^k)*(1 - x^k)^4), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(sqrt(7*n/3)*Pi) * 7^(5/4) / (32 * 3^(5/4) * n^(7/4)).

A318026 Expansion of Product_{k>=1} 1/((1 - x^k)(1 - x^(3k))).

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 16, 22, 33, 50, 70, 98, 143, 193, 266, 368, 493, 659, 892, 1170, 1543, 2035, 2642, 3422, 4448, 5694, 7294, 9334, 11839, 14982, 18968, 23812, 29868, 37410, 46598, 57924, 71953, 88913, 109728, 135212, 165991, 203407, 248986, 303706, 369939, 449967, 545820, 661038, 799629

Offset: 0

Ilya Gutkovskiy, Aug 13 2018

nonn

Convolution of A000041 and A035377.
Convolution of A000712 and A137569.
Convolution inverse of A030203.
Number of partitions of n if there are 2 kinds of parts that are multiples of 3.

			a(4) = 6 because we have [4], [3, 1], [3', 1], [2, 2], [2, 1, 1] and [1, 1, 1, 1].

Zakir Ahmed, Nayandeep Deka Baruah, Manosij Ghosh Dastidar, New congruences modulo 5 for the number of 2-color partitions, Journal of Number Theory, Volume 157, December 2015, Pages 184-198.
Index entries for sequences related to partitions

Cf. A000041, A000712, A002513, A030203, A030206, A035377, A112175, A137569, A318027, A318028.

a:=series(mul(1/((1-x^k)*(1-x^(3*k))),k=1..55),x=0,49): seq(coeff(a,x,n),n=0..48); # Paolo P. Lava, Apr 02 2019

nmax = 48; CoefficientList[Series[Product[1/((1 - x^k) (1 - x^(3 k))), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[1/(QPochhammer[x] QPochhammer[x^3]), {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[Sum[x^k (1 + x^k + 2 x^(2 k))/(k (1 - x^(3 k))), {k, 1, nmax}]], {x, 0, nmax}], x]
Table[Sum[PartitionsP[k] PartitionsP[n - 3 k], {k, 0, n/3}], {n, 0, 48}]

G.f.: exp(Sum_{k>=1} x^k*(1 + x^k + 2*x^(2*k))/(k*(1 - x^(3*k)))).

a(n) ~ exp(2*sqrt(2*n)*Pi/3) / (3 * 2^(5/4) * n^(5/4)). - Vaclav Kotesovec, Aug 14 2018

A318027 Expansion of Product_{k>=1} 1/((1 - x^k)(1 - x^(4k))).

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 13, 18, 29, 39, 57, 77, 112, 148, 205, 271, 372, 484, 647, 838, 1110, 1423, 1852, 2361, 3051, 3857, 4922, 6191, 7849, 9805, 12319, 15314, 19131, 23649, 29333, 36099, 44556, 54568, 66963, 81683, 99803, 121229, 147413, 178411, 216111, 260590, 314365, 377819, 454229

Offset: 0

Ilya Gutkovskiy, Aug 13 2018

nonn

Convolution of A000041 and A035444.
Convolution of A000712 and A082303.
Convolution inverse of A107034.
Number of partitions of n if there are 2 kinds of parts that are multiples of 4.

			a(5) = 8 because we have [5], [4, 1], [4', 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1] and [1, 1, 1, 1, 1].

Zakir Ahmed, Nayandeep Deka Baruah, Manosij Ghosh Dastidar, New congruences modulo 5 for the number of 2-color partitions, Journal of Number Theory, Volume 157, December 2015, Pages 184-198.
Index entries for sequences related to partitions

Cf. A000041, A000712, A002512 (self-convolution), A002513, A035444, A082303, A100853, A107034, A318026, A318028.

a:=series(mul(1/((1-x^k)*(1-x^(4*k))),k=1..55),x=0,49): seq(coeff(a,x,n),n=0..48); # Paolo P. Lava, Apr 02 2019

nmax = 48; CoefficientList[Series[Product[1/((1 - x^k) (1 - x^(4 k))), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[1/(QPochhammer[x] QPochhammer[x^4]), {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[Sum[x^k (1 + x^k + x^(2 k) + 2*x^(3 k))/(k (1 - x^(4 k))), {k, 1, nmax}]], {x, 0, nmax}], x]
Table[Sum[PartitionsP[k] PartitionsP[n - 4 k], {k, 0, n/4}], {n, 0, 48}]

G.f.: exp(Sum_{k>=1} x^k*(1 + x^k + x^(2*k) + 2*x^(3*k))/(k*(1 - x^(4*k)))).

a(n) ~ 5^(3/4) * exp(sqrt(5*n/6)*Pi) / (2^(13/4) * 3^(3/4) * n^(5/4)). - Vaclav Kotesovec, Aug 14 2018

A318028 Expansion of Product_{k>=1} 1/((1 - x^k)(1 - x^(5k))).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 17, 25, 35, 51, 69, 96, 129, 175, 235, 312, 410, 539, 700, 913, 1173, 1508, 1923, 2450, 3105, 3920, 4926, 6177, 7710, 9614, 11923, 14766, 18218, 22435, 27550, 33750, 41231, 50278, 61150, 74259, 89932, 108744, 131193, 158025, 189979, 227998, 273125, 326692

Offset: 0

Ilya Gutkovskiy, Aug 13 2018

nonn

Convolution of A000712 and A145466.
Convolution inverse of A030202.
Number of partitions of n if there are 2 kinds of parts that are multiples of 5.

			a(5) = 8 because we have [5], [5'], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1] and [1, 1, 1, 1, 1].

Zakir Ahmed, Nayandeep Deka Baruah, Manosij Ghosh Dastidar, New congruences modulo 5 for the number of 2-color partitions, Journal of Number Theory, Volume 157, December 2015, Pages 184-198.
Index entries for sequences related to partitions

Cf. A000712, A002513, A030202, A030205, A145466, A318026, A318027.

a:=series(mul(1/((1-x^k)*(1-x^(5*k))),k=1..55),x=0,49): seq(coeff(a,x,n),n=0..48); # Paolo P. Lava, Apr 02 2019

nmax = 48; CoefficientList[Series[Product[1/((1 - x^k) (1 - x^(5 k))), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[1/(QPochhammer[x] QPochhammer[x^5]), {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[Sum[x^k (1 + x^k + x^(2 k) + x^(3 k) + 2 x^(4 k))/(k (1 - x^(5 k))), {k, 1, nmax}]], {x, 0, nmax}], x]
Table[Sum[PartitionsP[k] PartitionsP[n - 5 k], {k, 0, n/5}], {n, 0, 48}]

G.f.: exp(Sum_{k>=1} x^k*(1 + x^k + x^(2*k) + x^(3*k) + 2*x^(4 k))/(k*(1 - x^(5*k)))).

a(n) ~ exp(2*Pi*sqrt(n/5)) / (4 * 5^(1/4) * n^(5/4)). - Vaclav Kotesovec, Aug 14 2018

A182031 Expansion of q^(-5/24) * (eta(q^3) * eta(q^6))^3 / (eta(q) * eta(q^2))^4 in powers of q.

Original entry on oeis.org

1, 4, 18, 53, 163, 414, 1059, 2431, 5553, 11844, 25013, 50391, 100362, 193136, 367371, 680705, 1247247, 2238408, 3975218, 6941384, 12003156, 20465599, 34581525, 57737205, 95601892, 156665029, 254777220, 410580026, 657015874

Offset: 0

Michael Somos, Apr 07 2012

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			1 + 4*x + 18*x^2 + 53*x^3 + 163*x^4 + 414*x^5 + 1059*x^6 + 2431*x^7 + ...
q^5 + 4*q^13 + 18*q^21 + 53*q^29 + 163*q^37 + 414*q^45 + 1059*q^53 + ...

H.-C. Chan, On the Andrews-Schur proof of the Rogers-Ramanujan identities, Ramanujan J. 23 (2010), no. 1-3, 417-431. see p. 430 Theorem 7.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002513.

eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-5/8)*(eta[q^3]*eta[q^6])^3/(eta[q]*eta[q^2])^4, {q, 0, 100}], q] (* G. C. Greubel, Apr 16 2018 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^6 + A))^3 / (eta(x + A) * eta(x^2 + A))^4, n))}

q='q+O('q^99); Vec((eta(q^3)*eta(q^6))^3/(eta(q)*eta(q^2))^4) \\ Altug Alkan, Apr 16 2018

Expansion of (psi(x^3) * phi(-x^3))^3 / (psi(x) * phi(-x))^4 in powers of x where phi(), psi() are Ramanujan theta functions.
Euler transform of period 6 sequence [ 4, 8, 1, 8, 4, 2, ...].
A002513(3*n + 2) = 3 * a(n).

A305168 Number of non-isomorphic graphs on 4n vertices whose edges are the union of two n-edge matchings.

Original entry on oeis.org

1, 3, 9, 23, 54, 118, 246, 489, 940, 1751, 3177, 5630, 9776, 16659, 27922, 46092, 75039, 120615, 191611, 301086, 468342, 721638, 1102113, 1669226, 2508429, 3741741, 5542532, 8155720, 11925654, 17334077, 25051940, 36009468, 51491111, 73263043, 103744575

Offset: 0

Yu Hin Au, Aug 17 2018

nonn

a(n) is also the number of partitions of 2n with two kinds of parts where all parts of the second kind are even. E.g., the a(2) = 9 such partitions are (2', 2'), (4'), (2,2'), (4), (1,1,2'), (3,1), (2,2), (2,1,1), (1,1,1,1). A bijection is to take each component in the graph whose edges are the union of two n-edge matchings, map each path of length p to a part p and each cycle (which must be even) of length p to a part p'.

			To see a(2)=9, observe that all graphs that are the union of two matchings of size n=2 are isomorphic to the union of S = {{1,2},{3,4}} and one of T=
  1. {{1,2}, {3,4}} --> (2',2')
  2. {{1,3}, {2,4}} --> (4')
  3. {{1,5}, {3,4}} --> (2,2')
  4. {{1,3}, {4,5}} --> (4)
  5. {{1,2}, {5,6}} --> (1,1,2')
  6. {{1,3}, {5,6}} --> (3,1)
  7. {{1,5}, {3,6}} --> (2,2)
  8. {{1,5}, {6,7}} --> (2,1,1)
  9. {{5,6}, {7,8}} --> (1,1,1,1)
Note that the partitions correspond to the bijection mentioned in the comments above.

Bisection (even part) of A002513.

Cf. A000041.

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(d*
      (2-irem(d, 2)), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> b(2*n):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 18 2018

a[n_] := Sum[PartitionsP[2k] PartitionsP[n-k], {k, 0, n}];
a /@ Range[0, 40] (* Jean-François Alcover, Nov 27 2020 *)

a(n) = sum(i=0, n, numbpart(2*i)*numbpart(n-i)); \\ Michel Marcus, Aug 18 2018

a(n) = [x^2n] (Product_{i>=1} 1/(1-x^i))*(Product_{j>=1} 1/(1-x^(2j))).
a(n) = Sum_{i=0..n} b(2i)*b(n-i) where b(n) is the number of partitions of n (A000041).
a(n) = A002513(2n). - Alois P. Heinz, Aug 18 2018

A319457 a(n) = [x^n] Product_{k>=1} 1/((1 - x^k)(1 - x^(2k)))^n.

Original entry on oeis.org

1, 1, 7, 31, 175, 931, 5209, 29114, 165087, 940828, 5396777, 31090962, 179832625, 1043516371, 6072302726, 35420582431, 207051636799, 1212583329959, 7113193757656, 41788933655049, 245831162935825, 1447891754747672, 8537111315442222, 50387162650271055, 297664212003582753

Offset: 0

Ilya Gutkovskiy, Sep 19 2018

nonn

Cf. A002513, A008485, A270919, A296043, A296044, A319455, A319456.

Table[SeriesCoefficient[Product[1/((1 - x^k) (1 - x^(2 k)))^n , {k, 1, n}], {x, 0, n}], {n, 0, 24}]
Table[SeriesCoefficient[1/(QPochhammer[x] QPochhammer[x^2])^n, {x, 0, n}], {n, 0, 24}]
Table[SeriesCoefficient[Exp[n Sum[(4 DivisorSigma[1, k] - DivisorSigma[1, 2 k]) x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 24}]

a(n) = [x^n] Product_{k>=1} (1 + x^k)^n/(1 - x^(2*k))^(2*n).

a(n) = [x^n] exp(n*Sum_{k>=1} (4*sigma(k) - sigma(2*k))*x^k/k).

A335602 Number of 3-regular cubic partitions of n.

Original entry on oeis.org

1, 1, 3, 3, 8, 9, 17, 20, 36, 43, 70, 84, 131, 158, 234, 284, 408, 495, 690, 837, 1143, 1385, 1852, 2241, 2952, 3565, 4626, 5574, 7150, 8595, 10903, 13074, 16434, 19656, 24494, 29223, 36146, 43016, 52836, 62722, 76572, 90675, 110063, 130021, 157014, 185049, 222388

Offset: 0

Chandrappa Shivashankar, Jun 15 2020

nonn

H.-C. Chan, Ramanujan's cubic continued fraction and a generalization of his "most beautiful identity", Int. J. Number Theory 6 (2010), 673--680.
H.-C. Chan, Ramanujan's cubic continued fraction and Ramanujan type congruences for a certain partition function, Int. J. Number Theory 6 (2010), 819--834.
S. Chern, Arithmetic Properties for Cubic Partition Pairs Modulo Powers of 3, Acta. Math. Sin.-English Ser. 2017 33: 1504.
D. S. Gireesh and M. S. Mahadeva Naika, General family of congruences modulo large powers of 3 for cubic partition pairs, New Zealand J. Math. 47 (2017), 43--56.

Cf. A002513, A335604.

nmax = 20; CoefficientList[Series[Product[(1 - x^(3*k)) * (1 - x^(6*k)) / ((1 - x^k) * (1 - x^(2*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 23 2020 *)

seq(n)={my(A=O(x*x^n)); Vec(eta(x^3 + A)*eta(x^6 + A)/(eta(x + A)*eta(x^2 + A)))} \\ Andrew Howroyd, Jul 29 2020

G.f.: (f_3(x)*f_6(x)) / (f_1(x)*f_2(x)) where f_k(x) = Product_{m>=1} (1 - x^(m*k)).

a(n) ~ exp(sqrt(2*n/3)*Pi) / (6^(5/4) * n^(3/4)). - Vaclav Kotesovec, Jun 23 2020

A335604 Number of 9-regular cubic partitions of n.

Original entry on oeis.org

1, 1, 3, 4, 9, 12, 23, 31, 54, 72, 117, 156, 242, 320, 477, 628, 909, 1188, 1676, 2178, 3012, 3888, 5283, 6780, 9079, 11582, 15309, 19424, 25389, 32040, 41462, 52063, 66780, 83448, 106182, 132084, 166862, 206660, 259359, 319896, 399069, 490272, 608234, 744444, 918864

Offset: 0

Chandrappa Shivashankar, Jun 15 2020

nonn

Hei-Chi Chan, Ramanujan's cubic continued fraction and a generalization of his "most beautiful identity", Int. J. Number Theory 6 (2010), 673--680.
Hei-Chi Chan, Ramanujan's cubic continued fraction and Ramanujan type congruences for a certain partition function, Int. J. Number Theory 6 (2010), 819--834.
S. Chern, Arithmetic Properties for Cubic Partition Pairs Modulo Powers of 3, Acta. Math. Sin.-English Ser. 2017 33: 1504.
Bernard L. S. Lin, Congruences modulo 27 for cubic partition pairs, J. Number Theory 171 (2017), 31--42.

Cf. A002513, A335602.

nmax = 50; CoefficientList[Series[Product[(1 - x^(9*k)) * (1 - x^(18*k)) / ((1 - x^k) * (1 - x^(2*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 23 2020 *)

seq(n)={my(A=O(x*x^n)); Vec(eta(x^9 + A)*eta(x^18 + A)/(eta(x + A)*eta(x^2 + A)))} \\ Andrew Howroyd, Jul 29 2020

G.f.: (f_9(x)*f_18(x)) / (f_1(x)*f_2(x)) where f_k(x) = Product_{m>=1} (1 - x^(m*k)).

a(n) ~ exp(2*sqrt(2*n)*Pi/3) / (2^(3/4) * 3^(5/2) * n^(3/4)). - Vaclav Kotesovec, Jun 23 2020

cp's OEIS Frontend

A029863 Expansion of Product_{k >= 1} 1/(1-x^k)^c(k), where c(1), c(2), ... = 2 3 2 3 2 3 2 3 ....

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A262380 Expansion of Product_{k>=1} 1/((1+x^k)*(1-x^k)^4).

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A318026 Expansion of Product_{k>=1} 1/((1 - x^k)*(1 - x^(3*k))).

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A318027 Expansion of Product_{k>=1} 1/((1 - x^k)*(1 - x^(4*k))).

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A318028 Expansion of Product_{k>=1} 1/((1 - x^k)*(1 - x^(5*k))).

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A182031 Expansion of q^(-5/24) * (eta(q^3) * eta(q^6))^3 / (eta(q) * eta(q^2))^4 in powers of q.

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A305168 Number of non-isomorphic graphs on 4n vertices whose edges are the union of two n-edge matchings.

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A318026 Expansion of Product_{k>=1} 1/((1 - x^k)(1 - x^(3k))).

A318027 Expansion of Product_{k>=1} 1/((1 - x^k)(1 - x^(4k))).

A318028 Expansion of Product_{k>=1} 1/((1 - x^k)(1 - x^(5k))).

A319457 a(n) = [x^n] Product_{k>=1} 1/((1 - x^k)(1 - x^(2k)))^n.