A003105 -id:A003105 - cp's OEIS frontend

A276527 Expansion of Product_{k>=1} 1 / (1 + x^k - x^(2*k)).

1, -1, 1, -3, 5, -8, 12, -21, 37, -59, 92, -153, 256, -409, 654, -1073, 1754, -2824, 4552, -7394, 12010, -19406, 31337, -50782, 82306, -133072, 215152, -348346, 563939, -912217, 1475604, -2388075, 3864808, -6252750, 10115987, -16369340, 26488326, -42857128

Offset: 0

Vaclav Kotesovec, Nov 16 2016

sign

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

Cf. A003105, A162891.

Cf. A000726, A109389, A263401.

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

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)) = 1.0964214808924344474065093...

A296164 a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 + x^(3*k)))^n.

Original entry on oeis.org

1, 1, 3, 10, 35, 131, 498, 1919, 7459, 29170, 114653, 452552, 1792754, 7124040, 28386081, 113372690, 453743907, 1819317153, 7306575042, 29386858821, 118348662525, 477188876405, 1926137365804, 7782398551661, 31472648050930, 127384123318906, 515978637418884

Offset: 0

Ilya Gutkovskiy, Dec 06 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Schur's Partition Theorem

Cf. A003105, A058484, A058539, A103262, A255526, A296163.

Table[SeriesCoefficient[Product[((1 + x^k)/(1 + x^(3 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[Product[1/((1 - x^(6 k - 1)) (1 - x^(6 k - 5)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
(* Calculation of constants {d,c}: *) With[{k = 3}, {1/r, Sqrt[QPochhammer[-1, (r*s)^k] / (2*Pi*(r^2*s*Derivative[0, 2][QPochhammer][-1, r*s] - k^2*(r*s)^(2*k) * Derivative[0, 2][QPochhammer][-1, (r*s)^k] - k*(1 + k)*(r*s)^k * Derivative[0, 1][QPochhammer][-1, (r*s)^k]))]} /. FindRoot[{s == QPochhammer[-1, r*s]/QPochhammer[-1, (r*s)^k], QPochhammer[-1, (r*s)^k] + k*(r*s)^k*Derivative[0, 1][QPochhammer][-1, (r*s)^k] == r*Derivative[0, 1][QPochhammer][-1, r*s]}, {r, 1/4}, {s, 2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)

a(n) = [x^n] Product_{k>=1} 1/((1 - x^(6*k-1))*(1 - x^(6*k-5)))^n.

a(n) ~ c * d^n / sqrt(n), where d = 4.129321588075726742506... and c = 0.25764349816429874321... - Vaclav Kotesovec, May 18 2018

A304632 Expansion of (1/(1 - x))* Product_{k>=1} (1 + x^k)/(1 + x^(3*k)).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 12, 15, 18, 22, 27, 33, 40, 48, 57, 67, 79, 93, 109, 127, 147, 170, 196, 226, 260, 298, 340, 387, 440, 500, 567, 641, 723, 814, 916, 1030, 1156, 1295, 1448, 1617, 1804, 2011, 2239, 2489, 2763, 3064, 3395, 3759, 4158, 4594, 5070, 5590, 6159, 6781, 7460, 8199, 9003

Offset: 0

Ilya Gutkovskiy, May 15 2018

nonn

Partial sums of A003105.

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
Eric Weisstein's World of Mathematics, Schur's Partition Theorem
Index entries for sequences related to partitions

Cf. A000070, A003105, A036469, A304630, A304631.

nmax = 57; CoefficientList[Series[1/(1 - x) Product[(1 + x^k)/(1 + x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 57; CoefficientList[Series[1/(1 - x) Product[1/((1 - x^(6 k + 1)) (1 - x^(6 k + 5))), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 57; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^k + x^(2 k)), {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: (1/(1 - x))*Product_{k>=0} 1/((1 - x^(6*k+1))*(1 - x^(6*k+5))).
G.f.: (1/(1 - x))*Product_{k>=0} 1/(1 - x^k + x^(2*k)).
a(n) ~ exp(sqrt(2*n)*Pi/3) * sqrt(3) / (Pi * 2^(3/4) * n^(1/4)). - Vaclav Kotesovec, May 19 2018

A332309 Number of compositions (ordered partitions) of n into distinct parts where no part is a multiple of 3.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 4, 9, 9, 6, 15, 17, 38, 29, 53, 70, 65, 91, 150, 229, 277, 236, 439, 489, 514, 897, 993, 1632, 1521, 2339, 2972, 3257, 4121, 5992, 5303, 7729, 10932, 15157, 17653, 18398, 26305, 31683, 34408, 51885, 58173, 61098, 90519, 101249, 143402, 156905

Offset: 0

Ilya Gutkovskiy, Feb 09 2020

nonn

			a(9) = 6 because we have [8, 1], [7, 2], [5, 4], [4, 5], [2, 7] and [1, 8].

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Index entries for sequences related to compositions

Cf. A000726, A003105, A032020, A032021, A332310, A332311.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$2, 0):
seq(a(n), n=0..55);  # Alois P. Heinz, Feb 09 2020

b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0, If[n == 0, p!, Sum[b[n - i j, i - 1, p + j], {j, 0, Min[Mod[i, 3], 1, n/i]}]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 55] (* Jean-François Alcover, Nov 09 2020, after Alois P. Heinz *)

A101230 Number of partitions of 2n in which both odd parts and parts that are multiples of 3 occur with even multiplicities. There is no restriction on the other even parts.

Original entry on oeis.org

1, 2, 4, 7, 12, 20, 32, 50, 76, 113, 166, 240, 343, 484, 676, 935, 1282, 1744, 2355, 3158, 4208, 5573, 7340, 9616, 12536, 16266, 21012, 27028, 34628, 44196, 56204, 71226, 89964, 113270, 142180, 177948, 222089, 276430, 343172, 424959, 524966

Offset: 0

Noureddine Chair, Dec 16 2004

nonn

Note that if a partition of n has odd parts occur with even multiplicities then n must be even. This is the reason for only looking at partitions of 2n. - Michael Somos, Mar 04 2012

			a(8)=12 because 8 = 4+4 = 4+2+2 = 4+2+1+1 = 4+1+1+1+1 = 3+3+2 = 3+3+1+1 = 2+2+2+2 = 2+2+2+1+1 = 2+2+1+1+1+1 = 2+1+1+1+1+1+1 = 1+1+1+1+1+1+1+1.
1 + 2*x + 4*x^2 + 7*x^3 + 12*x^4 + 20*x^5 + 32*x^6 + 50*x^7 + 76*x^8 + 113*x^9 + ...
1/q + 2*q^7 + 4*q^15 + 7*q^23 + 12*q^31 + 20*q^39 + 32*q^47 + 50*q^55 + 76*q^63 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.

Cf. A000041, A003105, A015128, A089812, A098151.

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

nmax = 50; CoefficientList[Series[Product[(1+x^(3*k-1))*(1+x^(3*k-2)) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 01 2015 *)

{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)^2 * eta(x^6 + A)), n))} /* Michael Somos, Mar 04 2012 */

G.f.: product_{k>0}(1+x^k)/((1-x^k)(1+x^(3k)))= Theta_4(0, x^3)/theta(0, x)1/product_{k>0}(1-x^(3k)).
Euler transform of period 6 sequence [2, 1, 1, 1, 2, 1, ...]. - Vladeta Jovovic, Dec 17 2004
Expansion of q^(1/8) * eta(q^2) * eta(q^3) / (eta(q)^2 * eta(q^6)) in powers of q. - Michael Somos, Mar 04 2012
Convolution inverse of A089812. - Michael Somos, Mar 04 2012
Convolution product of A000041 and A003105. - Michael Somos, Mar 04 2012
a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (6*n). - Vaclav Kotesovec, Sep 01 2015

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

A227398 Expansion of chi(x^3) / chi(x) in powers of x where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, 1, -1, 1, -2, 2, -3, 3, -3, 4, -5, 6, -7, 8, -9, 10, -12, 14, -16, 18, -20, 23, -26, 30, -34, 38, -42, 47, -53, 60, -67, 74, -82, 91, -102, 114, -126, 139, -153, 169, -187, 207, -228, 250, -274, 301, -331, 364, -399, 436, -476, 520, -569, 622, -679

Offset: 0

Michael Somos, Sep 20 2013

sign

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

			G.f. = 1 - x + x^2 - x^3 + x^4 - 2*x^5 + 2*x^6 - 3*x^7 + 3*x^8 - 3*x^9 + ...
G.f. = 1/q - q^11 + q^23 - q^35 + q^47 - 2*q^59 + 2*q^71 - 3*q^83 + 3*q^95 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A003105, A098884.

a[ n_] := SeriesCoefficient[ 1 / Product[ 1 - (-x)^k + x^(2 k), {k, n}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k + x^(2 k), {k, 1, n, 2}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, 3, n, 6}] / Product[ 1 + x^k, {k, 1, n, 2}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ 1 / (Product[ 1 + x^k, {k, 1, n, 6}] Product[ 1 + x^k, {k, 5, n, 6}]), {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ 1 + (-x)^k, {k, 1, n, 3}] Product[ 1 + (-x)^k, {k, 2, n, 3}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^3, x^6] / QPochhammer[ -x, x^2], {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2 / (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A)), n))};

{a(n) = my(A, m); if( n<0, 0, A = x * O(x^n); m = sqrtint(3*n + 1); polcoeff( eta(x^6 + A) / sum(k= -((m-1)\3), (m+1)\3, x^(k * (3*k - 2)), A), n))};

Expansion of f(-x^6) / f(x, x^5) in powers of x where f(,) is Ramanujan's general theta function.
Expansion of q^(1/12) * eta(q) * eta(q^4) * eta(q^6)^2 / (eta(q^2)^2 * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [-1, 1, 0, 0, -1, 0, -1, 0, 0, 1, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A098884.
G.f.: 1 / Product_{k>0} (1 - (-x)^k + x^(2*k)).
G.f.: Product_{k>0} (1 - x^(2*k - 1) + x^(4*k - 2)).
G.f.: Product_{k>0} ((1 + x^(6*k - 3)) / (1 + x^(2*k - 1))).
G.f.: 1 / Product_{k>0} ((1 + x^(6*k - 1)) * (1 + x^(6*k - 5))).
G.f.: Product_{k>0} (1  + (-x)^(3*k - 1)) * (1 + (-x)^(3*k - 2)).
G.f.: (Sum_{k in Z} (-1)^k * x^(3*k * (3*k-1))) / (Sum_{k in Z} x^(k * (3*k - 2))).
a(n) = (-1)^n * A003105(n). Convolution inverse of A098884.

A293302 E.g.f.: Product_{m>0} 1/(1 - x^m + x^(2*m)/2!).

Original entry on oeis.org

1, 1, 3, 12, 66, 450, 3510, 32760, 335160, 3832920, 48648600, 673596000, 9961736400, 161026866000, 2775402630000, 50713246584000, 987048958896000, 20331148966128000, 440625863806128000, 10057578887708352000, 240218186856167520000, 6010719623406257760000

Offset: 0

Seiichi Manyama, Oct 05 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..444

Column k=2 of A293301.

Cf. A003105.

nmax = 25; CoefficientList[Series[1/Product[1 - x^k + x^(2*k)/2, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 05 2017 *)

my(x = 'x + O('x^40)); Vec(serlaplace(prod(m=1, 40, 1/(1 - x^m + x^(2*m)/2!)))) \\ Michel Marcus, Oct 05 2017

a(n) ~ (5*Pi^2/3 - 4*log(2)^2)^(1/4) * n^(n - 1/4) / (4*exp(n - sqrt((5*Pi^2/12 - log(2)^2)*n))). - Vaclav Kotesovec, Oct 07 2024

A260183 Expansion of f(x, x^2) * f(x^4, x^8) / f(-x^3, -x^6)^2 in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 10, 14, 18, 23, 30, 38, 47, 60, 74, 91, 114, 139, 169, 207, 250, 301, 364, 436, 520, 622, 739, 875, 1038, 1224, 1439, 1694, 1985, 2321, 2714, 3162, 3677, 4275, 4956, 5735, 6634, 7655, 8819, 10155, 11669, 13389, 15354, 17575, 20091

Offset: 0

Michael Somos, Nov 10 2015

nonn

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 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 10*x^8 + 14*x^9 + ...
G.f. = 1/q + q + q^3 + 2*q^5 + 3*q^7 + 4*q^9 + 6*q^11 + 8*q^13 + 10*q^15 + ...

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. A003105.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^12] QPochhammer[ x^10, x^12] QPochhammer[ x^12, x^24] QPochhammer[ x^8] / QPochhammer[x], {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^24 + A)), n))};

Expansion of q^(1/2) * eta(q^2) * eta(q^8) * eta(q^12)^2 / (eta(q) * eta(q^4) * eta(q^6) * eta(q^24)) in powers of q.
Euler transform of period 24 sequence [ 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, ...].
a(n) = A003105(2*n + 1).

A284092 Number of partitions of n into distinct parts 8k+1 or 8k+7.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 2, 2, 1, 0, 0, 0, 1, 2, 3, 3, 2, 1, 0, 0, 1, 3, 5, 5, 3, 1, 0, 0, 2, 5, 7, 7, 5, 2, 0, 1, 3, 7, 11, 11, 7, 3, 1, 1, 5, 11, 15, 15, 11, 5, 1, 2, 7, 15, 22, 22, 15, 7, 2, 3, 11, 22, 30, 30, 22, 11, 4, 5, 15, 30, 42, 42

Offset: 0

Seiichi Manyama, Mar 20 2017

nonn

Convolution of A284093 and A284095.

Vaclav Kotesovec, Table of n, a(n) for n = 0..20000

Cf. A035453, A284093, A284095.

Cf. Product_{k>0} (1 + x^(m*k - 1)) * (1 + x^(m*k - m + 1)): A003105 (m=3), A000700 (m=4), A203776 (m=5), A098884 (m=6), A281459 (m=7), this sequence (m=8).

CoefficientList[Series[Product[(1 + x^(8*k - 1)) * (1 + x^(8*k - 7)) , {k, 1, 81}], {x, 0, 81}], x] (* Indranil Ghosh, Mar 20 2017 *)

Vec(prod(k=1, 81, (1 + x^(8*k - 1)) * (1 + x^(8*k - 7))) + O(x^82)) \\ Indranil Ghosh, Mar 20 2017

G.f.: Product_{k>0} (1 + x^(8*k - 1)) * (1 + x^(8*k - 7)).

a(n) ~ exp(sqrt(n/3)*Pi/2) / (4*3^(1/4)*n^(3/4)) * (1 + (11*Pi/(192*sqrt(3)) - 3*sqrt(3)/(4*Pi))/sqrt(n)). - Vaclav Kotesovec, Mar 20 2017

cp's OEIS Frontend

A276527 Expansion of Product_{k>=1} 1 / (1 + x^k - x^(2*k)).

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A296164 a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 + x^(3*k)))^n.

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A304632 Expansion of (1/(1 - x))* Product_{k>=1} (1 + x^k)/(1 + x^(3*k)).

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A332309 Number of compositions (ordered partitions) of n into distinct parts where no part is a multiple of 3.

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A101230 Number of partitions of 2n in which both odd parts and parts that are multiples of 3 occur with even multiplicities. There is no restriction on the other even parts.

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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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A227398 Expansion of chi(x^3) / chi(x) in powers of x where chi() is a Ramanujan theta function.

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A293302 E.g.f.: Product_{m>0} 1/(1 - x^m + x^(2*m)/2!).

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