A080054 -id:A080054 - cp's OEIS frontend

A261649 Expansion of Product_{k>=0} ((1+x^(3k+1))/(1-x^(3k+1)))^2.

1, 4, 8, 12, 20, 36, 56, 80, 120, 180, 252, 348, 492, 680, 912, 1228, 1652, 2180, 2856, 3744, 4860, 6256, 8044, 10284, 13048, 16520, 20848, 26140, 32672, 40756, 50596, 62576, 77256, 95060, 116540, 142592, 174036, 211736, 257056, 311448, 376332, 453764, 546160

Offset: 0

Vaclav Kotesovec, Aug 28 2015

nonn

Cf. A015128, A156616, A080054, A007096, A261610, A261651.

nmax=60; CoefficientList[Series[Product[((1+x^(3*k+1))/(1-x^(3*k+1)))^2,{k,0,nmax}],{x,0,nmax}],x]

a(n) ~ exp(Pi*sqrt(2*n/3)) * Gamma(1/3)^2 / (2^(7/4) * 3^(5/12) * Pi^(4/3) * n^(7/12)).

A261651 Expansion of Product_{k>=0} ((1+x^(3k+1))/(1-x^(3k+1)))^3.

Original entry on oeis.org

1, 6, 18, 38, 72, 138, 254, 432, 708, 1154, 1836, 2826, 4288, 6456, 9552, 13902, 20070, 28722, 40614, 56916, 79242, 109448, 149988, 204318, 276672, 372288, 498264, 663602, 879252, 1159470, 1522564, 1990788, 2592162, 3362638, 4346244, 5597100, 7183792, 9191004

Offset: 0

Vaclav Kotesovec, Aug 28 2015

nonn

Cf. A015128, A156616, A080054, A007096, A261610, A261649.

nmax=60; CoefficientList[Series[Product[((1+x^(3*k+1))/(1-x^(3*k+1)))^3,{k,0,nmax}],{x,0,nmax}],x]

a(n) ~ exp(Pi*sqrt(n)) * Gamma(1/3)^3 / (4 * Pi^2 * sqrt(3*n)).

A261652 Expansion of Product_{k>=0} ((1+x^(4k+1))/(1-x^(4k+1)))^3.

Original entry on oeis.org

1, 6, 18, 38, 66, 108, 182, 306, 486, 728, 1068, 1578, 2318, 3312, 4614, 6388, 8862, 12192, 16488, 22038, 29400, 39156, 51702, 67554, 87810, 113982, 147384, 189200, 241446, 307356, 390408, 493662, 621006, 778712, 974628, 1216284, 1511756, 1872840, 2315538

Offset: 0

Vaclav Kotesovec, Aug 28 2015

nonn

In general, if j > 0, a > 0, b > 0, GCD(a,b) = 1 and g.f. = Product_{k>=0} ((1 + x^(a*k+b))/(1 - x^(a*k+b)))^j, then a(n) ~ Gamma(b/a)^j * 2^(j/2 - 3/2 - 2*b*j/a) * a^(-j/4 - 1/4 + b*j/(2*a)) * exp(Pi*sqrt(j*n/a)) * j^(1/4 - j/4 + b*j/(2*a)) * Pi^(b*j/a - j) * n^(j/4 - 3/4 - b*j/(2*a)).

Cf. A015128 (a=1, b=1, j=1), A156616.
Cf. A080054 (a=2, b=1, j=1), A007096 (a=2, b=1, j=2), A261647 (a=2, b=1, j=3), A014969 (a=2, b=1, j=4), A261648 (a=2, b=1, j=5), A014970 (a=2, b=1, j=6), A014972 (a=2, b=1, j=8), A103261 (a=2, b=1, j=10).
Cf. A261610 (a=3, b=1, j=1), A261649 (a=3, b=1, j=2), A261651 (a=3, b=1, j=3).
Cf. A261611 (a=4, b=1, j=1), A261650 (a=4, b=1, j=2), A261652 (a=4, b=1, j=3).

nmax=60; CoefficientList[Series[Product[((1+x^(4*k+1))/(1-x^(4*k+1)))^3,{k,0,nmax}],{x,0,nmax}],x]

a(n) ~ exp(Pi*sqrt(3*n)/2) * 2^(1/4) * Gamma(1/4)^3 / (8 * 3^(1/8) * Pi^(9/4) * n^(3/8)).

A103260 Number of partitions of 2n prime to 3 with all odd parts occurring with multiplicity 2. The even parts occur with multiplicity 1.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 6, 8, 10, 10, 12, 16, 22, 28, 32, 36, 42, 52, 66, 80, 92, 104, 120, 144, 174, 206, 236, 266, 304, 356, 420, 488, 554, 624, 708, 816, 946, 1084, 1224, 1372, 1548, 1764, 2016, 2288, 2568, 2868, 3216, 3632, 4110, 4626, 5166, 5748, 6412, 7188

Offset: 0

Noureddine Chair, Feb 15 2005

Convolution of A098884 and A003105. [corrected by Vaclav Kotesovec, Feb 07 2021]

Also equal to the number of overpartitions of n into parts congruent to 1 or 5 modulo 6. - Jeremy Lovejoy, Nov 28 2024

			E.g. a(7)=8 because 14=10+4=10+2+1+1=8+4+2=8+4+1+1=7+7=5+5+4=5+5+2+1+1.

Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A003105, A098884, A080054, A001082, A098151, A015128.

series(product(((1+x^(6*k-1))*(1+x^(6*k-5)))/((1-x^(6*k-1))*(1-x^(6*k-5))),k=1..100),x=0,100);
# alternative program:
with(gfun): series( add(x^(n*(3*n-2)), n = -6..6)/add((-1)^n*x^(n*(3*n-2)), n = -6..6), x, 100): seriestolist(%); # Peter Bala, Feb 05 2021

nmax = 50; CoefficientList[Series[Product[((1+x^(6*k-1))*(1+x^(6*k-5)))/((1-x^(6*k-1))*(1-x^(6*k-5))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 01 2015 *)

G.f.: (Theta_4(0, x^2)*theta_4(0, x^3))/(theta_4(0, x)*theta_4(0, x^(6))) = Product_{k>0}((1+x^(6*k-1))*(1+x^(6*k-5)))/((1-x^(6*k-1))*(1-x^(6*k-5))).
Euler transform of period 12 sequence [2, -1, 0, 0, 2, 0, 2, 0, 0, -1, 2, 0, ...]. - Vladeta Jovovic, Feb 17 2005
a(n) ~ exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 01 2015
G.f.: f(x,x^5)/f(-x,-x^5) = ( Sum_{n = -oo..oo} x^(n*(3*n-2)) )/( Sum_{n = -oo..oo} (-1)^n*x^(n*(3*n-2)) ), where f(a,b) = Sum_{n = -oo..oo} a^(n*(n+1)/2)*b^(n*(n-1)/2) is Ramanujan's 2-variable theta function. Cf. A080054 and A098151. - Peter Bala, Feb 05 2021

Example corrected by Vaclav Kotesovec, Sep 01 2015

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.

Original entry on oeis.org

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

A207944 Expansion of Product_{i>=1} (1 + x^(2i + 1))/(1 - x^(2i + 1)).

Original entry on oeis.org

1, 0, 0, 2, 0, 2, 2, 2, 4, 4, 6, 6, 10, 10, 14, 18, 20, 26, 32, 38, 46, 58, 66, 82, 98, 116, 138, 166, 194, 230, 274, 318, 376, 442, 514, 602, 704, 814, 950, 1102, 1274, 1474, 1706, 1962, 2262, 2606, 2986, 3430, 3934, 4496, 5144, 5878, 6698, 7638, 8698, 9886

Offset: 0

Roger L. Bagula, Feb 21 2012

nonn

George E. Andrews, Number Theory, Dover Publications, N.Y., 1971, 164-165.
Samuel I. Goldberg, Curvature and Homology, Dover, New York, 1998, 144.

Shi-Chao Chen, On the number of overpartitions into odd parts, Discrete Math. 325 (2014), 32--37. MR3181230. The g.f. occurs on page 32, by accident (the product there should really start at n=0, not 1, in order to give A080054!). - _N. J. A. Sloane_, Apr 24 2014

Cf. A015128, A080054, A142724.

nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k + 1))/(1 - x^(2*k + 1)), {k, 1, nmax}], {x, 0, nmax}], x] (* fixed by Vaclav Kotesovec, Apr 13 2017 *)

a(n) ~ exp(sqrt(n/2)*Pi) * Pi / (2^(19/4) * n^(5/4)). - Vaclav Kotesovec, Apr 13 2017

A261156 Expansion of chi(q) * chi(-q^9) / (chi(-q) * chi(q^9)) in powers of q where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 2, 4, 6, 8, 12, 16, 22, 28, 36, 48, 60, 76, 96, 120, 150, 184, 228, 280, 340, 416, 504, 608, 732, 878, 1052, 1252, 1488, 1768, 2088, 2464, 2902, 3408, 3996, 4672, 5460, 6364, 7400, 8600, 9972, 11544, 13344, 15400, 17752, 20424, 23472, 26944, 30876, 35346

Offset: 0

Michael Somos, Aug 10 2015

nonn

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

			G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 6*x^4 + 8*x^5 + 12*x^6 + 16*x^7 + 22*x^8 + ...

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. A080054, A123629, A186924, A187100, A212484, A233693, A260215.

a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2] QPochhammer[ -q, q] QPochhammer[ q^9, q^18] QPochhammer[ q^9, -q^9], {q, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^9 + A)^2 * eta(x^36 + A) / (eta(x + A)^2 * eta(x^4 + A) * eta(x^18 + A)^3), n))};

Expansion of eta(q^2)^3 * eta(q^9)^2 * eta(q^36) / (eta(q)^2 * eta(q^4) * eta(q^18)^3) in powers of q.
G.f. A(x) = B(x) / B(x^9) where B(x) is the g.f. of A080054.
Euler transform of period 36 sequence [ 2, -1, 2, 0, 2, -1, 2, 0, 0, -1, 2, 0, 2, -1, 2, 0, 2, 0, 2, 0, 2, -1, 2, 0, 2, -1, 0, 0, 2, -1, 2, 0, 2, -1, 2, 0, ...].
a(n) = 2 * A233693(n) unless n=0.  a(2*n) = 2 * A123629(n) = 2 * A212484(n) unless n=0.
a(3*n) = A186924(n). a(3*n) = 4 * A187100(n) unless n=0.
a(n) = (-1)^n * A260215(n). - Michael Somos, Aug 14 2015
a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

A261650 Expansion of Product_{k>=0} ((1+x^(4k+1))/(1-x^(4k+1)))^2.

Original entry on oeis.org

1, 4, 8, 12, 16, 24, 40, 60, 80, 104, 144, 204, 272, 344, 440, 584, 768, 968, 1200, 1516, 1936, 2424, 2968, 3644, 4528, 5596, 6800, 8216, 10000, 12184, 14688, 17564, 21056, 25320, 30272, 35912, 42576, 50616, 60024, 70728, 83136, 97896, 115200, 134924, 157504

Offset: 0

Vaclav Kotesovec, Aug 28 2015

nonn

Cf. A015128, A156616, A080054, A007096, A261611, A261652.

nmax=60; CoefficientList[Series[Product[((1+x^(4*k+1))/(1-x^(4*k+1)))^2,{k,0,nmax}],{x,0,nmax}],x]

a(n) ~ exp(Pi*sqrt(n/2)) * 2^(3/2) * Gamma(1/4)^2 / (16 * Pi^(3/2) * sqrt(n)).

A289522 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=0} ((1 + x^(2j+1))/(1 - x^(2j+1)))^k.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 2, 0, 1, 6, 8, 4, 0, 1, 8, 18, 16, 6, 0, 1, 10, 32, 44, 32, 8, 0, 1, 12, 50, 96, 102, 56, 12, 0, 1, 14, 72, 180, 256, 216, 96, 16, 0, 1, 16, 98, 304, 550, 624, 428, 160, 22, 0, 1, 18, 128, 476, 1056, 1512, 1408, 816, 256, 30, 0, 1, 20, 162, 704, 1862, 3240, 3820, 3008, 1494, 404, 40, 0

Offset: 0

Ilya Gutkovskiy, Jul 07 2017

			Square array begins:
1,  1,   1,    1,    1,     1,  ...
0,  2,   4,    6,    8,    10,  ...
0,  2,   8,   18,   32,    50,  ...
0,  4,  16,   44,   96,   180,  ...
0,  6,  32,  102,  256,   550,  ...
0,  8,  56,  216,  624,  1512,  ...

Columns k=0-6 give: A000007, A080054, A007096, A261647, A014969, A261648, A014970.
Rows n=0-3 give: A000012, A005843, A001105, A217873.
Main diagonal gives A291697.

Table[Function[k, SeriesCoefficient[Product[((1 + x^(2 i + 1))/(1 - x^(2 i + 1)))^k, {i, 0, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[(QPochhammer[-x, x^2]/QPochhammer[x, x^2])^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=0} ((1 + x^(2*j+1))/(1 - x^(2*j+1)))^k.
G.f. of column 2k: (theta_3(x)/theta_4(x))^k, where theta_() is the Jacobi theta function.
For asymptotics of column k see comment from Vaclav Kotesovec in A261648.

A292563 Expansion of Product_{k>=1} (1 + x^((2k-1)^3)) / (1 - x^((2k-1)^3)).

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 8, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 0

Vaclav Kotesovec, Sep 19 2017

nonn

Convolution of A292547 and A287091.

In general, if m > 0 and g.f. = Product_{k>=1} (1 + x^((2*k-1)^m)) / (1 - x^((2*k-1)^m)), then a(n) ~ exp((m+1) * ((2^(1 + 1/m)-1) * Gamma(1/m) * Zeta(1 + 1/m)/m^2)^(m/(m+1)) * n^(1/(m+1)) / 2) * ((2^(1 + 1/m)-1) * Gamma(1/m) * Zeta(1 + 1/m))^(m/(2*(m+1))) / (sqrt(Pi*(m+1)) * 2^(m/2 + 1) * m^((m-1)/(2*(m+1))) * n^((2*m+1)/(2*(m+1)))).

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

Cf. A080054 (m=1), A104274 (m=2).

Cf. A287091, A292547.

nmax = 100; CoefficientList[Series[Product[(1 + x^((2*k-1)^3)) / (1 - x^((2*k-1)^3)), {k, 1, Floor[nmax^(1/3)/2] + 1}], {x, 0, nmax}], x]

a(n) ~ exp(2 * ((2^(4/3)-1) * Gamma(1/3) * Zeta(4/3))^(3/4) * n^(1/4) / 3^(3/2)) * ((2^(4/3)-1) * Gamma(1/3) * Zeta(4/3))^(3/8) / (2^(7/2) * 3^(1/4) * sqrt(Pi) * n^(7/8)).

cp's OEIS Frontend

A261649 Expansion of Product_{k>=0} ((1+x^(3*k+1))/(1-x^(3*k+1)))^2.

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A261651 Expansion of Product_{k>=0} ((1+x^(3*k+1))/(1-x^(3*k+1)))^3.

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A261652 Expansion of Product_{k>=0} ((1+x^(4*k+1))/(1-x^(4*k+1)))^3.

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A103260 Number of partitions of 2n prime to 3 with all odd parts occurring with multiplicity 2. The even parts occur with multiplicity 1.

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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.

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A207944 Expansion of Product_{i>=1} (1 + x^(2*i + 1))/(1 - x^(2*i + 1)).

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A261156 Expansion of chi(q) * chi(-q^9) / (chi(-q) * chi(q^9)) in powers of q where chi() is a Ramanujan theta function.

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A261650 Expansion of Product_{k>=0} ((1+x^(4*k+1))/(1-x^(4*k+1)))^2.

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A261649 Expansion of Product_{k>=0} ((1+x^(3k+1))/(1-x^(3k+1)))^2.

A261651 Expansion of Product_{k>=0} ((1+x^(3k+1))/(1-x^(3k+1)))^3.

A261652 Expansion of Product_{k>=0} ((1+x^(4k+1))/(1-x^(4k+1)))^3.

A207944 Expansion of Product_{i>=1} (1 + x^(2i + 1))/(1 - x^(2i + 1)).

A261650 Expansion of Product_{k>=0} ((1+x^(4k+1))/(1-x^(4k+1)))^2.

A289522 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=0} ((1 + x^(2j+1))/(1 - x^(2j+1)))^k.

A292563 Expansion of Product_{k>=1} (1 + x^((2k-1)^3)) / (1 - x^((2k-1)^3)).