A273845 -id:A273845 - cp's OEIS frontend

A005882 Theta series of planar hexagonal lattice (A2) with respect to deep hole.

3, 3, 6, 0, 6, 3, 6, 0, 3, 6, 6, 0, 6, 0, 6, 0, 9, 6, 0, 0, 6, 3, 6, 0, 6, 6, 6, 0, 0, 0, 12, 0, 6, 3, 6, 0, 6, 6, 0, 0, 3, 6, 6, 0, 12, 0, 6, 0, 0, 6, 6, 0, 6, 0, 6, 0, 9, 6, 6, 0, 6, 0, 0, 0, 6, 9, 6, 0, 0, 6, 6, 0, 12, 0, 6, 0, 6, 0, 0, 0, 6, 6, 12, 0, 0, 3, 12, 0, 0, 6, 6, 0, 6, 0, 6, 0, 3, 6, 0, 0, 12

Offset: 0

N. J. A. Sloane

nonn

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
On page 111 of Conway and Sloane is "If the origin is moved to a deep hole the theta series is Theta_{hex+[1]}(z) = theta_2(z) psi_6(3z) + theta_3(z) psi_3(3z) = 3 q^{1/3} + 3 q^{4/3} + 6 q^{7/3} + 6 q^{13/3} + ... (63)" where the psi_k() for integer k is defined on page 103 equation (11) as psi_k(z) = e^{Pi i/z^2} theta_3(Pi z/k | z) = Sum_{m in Z} q^{(m + 1/k)^2}. - Michael Somos, Sep 10 2018

			G.f. = 3 + 3*x + 6*x^2 + 6*x^4 + 3*x^5 + 6*x^6 + 3*x^8 + 6*x^9 + 6*x^10 + ...
G.f. = 3*q + 3*q^4 + 6*q^7 + 6*q^13 + 3*q^16 + 6*q^19 + 3*q^25 + 6*q^28 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701. MR1010408 (91e:33012) see page 695.
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015. [Note that a later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication which is next in this list.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Essentially same as A033685 and A033687.

Cf. A113062, A273845.

Basis( ModularForms( Gamma1(9), 1), 302)[2] * 3; /* Michael Somos, Jul 19 2014 */

a[ n_] := SeriesCoefficient[ 3 QPochhammer[ q^3]^3 / QPochhammer[ q], {q, 0, n}]; (* Michael Somos, Jul 19 2014 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( 3 * eta(x^3 + A)^3 / eta(x + A), n))}; /* Michael Somos, Aug 15 2006 */

Expansion of q^(-1/3) * 3 * eta(q^3)^3 / eta(q) in powers of q.
Expansion of q^(-1/3) * c(q) in powers of q where c(q) is the third cubic AGM theta function.
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^3 + 2*u*w^2 - u^2*w. - Michael Somos, Aug 15 2006
G.f.: 3 Product_{k>0} (1-q^(3k))^3/(1-q^k).
G.f.: Sum_{u,v in Z} x^(u*u + u*v + v*v + u + v). - Michael Somos, Jul 19 2014
a(n) = 3 * A033687(n). a(n) = A113062(3*n + 1) = A033685(3*n + 1).
Expansion of 2 * psi(x^2) * f(x^2, x^4) + phi(x) * f(x^1, x^5) in powers of x where phi(), psi() are Ramanujan theta functions and f(, ) is Ramanujan's general theta function. - Michael Somos, Sep 07 2018
Sum_{k=1..n} a(k) ~ 2*Pi*n/sqrt(3). - Vaclav Kotesovec, Dec 17 2022

A277212 Expansion of Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^5 in powers of x.

Original entry on oeis.org

1, 5, 20, 65, 190, 505, 1260, 2970, 6700, 14535, 30520, 62235, 123720, 240340, 457380, 854190, 1568230, 2834120, 5048140, 8871450, 15396690, 26410860, 44811440, 75254240, 125162100, 206275505, 337032360, 546183425, 878270360, 1401857550, 2221862260

Offset: 0

Seiichi Manyama, Nov 07 2016

nonn

In general, for fixed m > 1, if g.f. = Product_{k>=1} (1 - x^(m*k))/(1 - x^k)^m, then a(n, m) ~ exp(Pi*sqrt(2*n*(m-1/m)/3)) * (m^2 - 1)^(m/4) / (2^(3*m/4 + 1/2) * 3^(m/4) * m^(m/4 + 1/2) * n^(m/4 + 1/2)). - Vaclav Kotesovec, Nov 10 2016

			G.f.: 1 + 5*x + 20*x^2 + 65*x^3 + 190*x^4 + 505*x^5 + 1260*x^6 + ...

Robert Israel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol

Cf. Expansion of Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k in powers of x: A015128 (k=2), A273845 (k=3), A274327 (k=4), this sequence (k=5), A160539 (k=7).

Cf. A109064.

N:= 100: # to get a(0)..a(N)
S:= series(mul((1-x^(5*n))/(1-x^n)^5,n=1..N),x,N+1):
seq(coeff(S,x,n),n=0..N); # Robert Israel, Nov 09 2016

nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))/(1 - x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *)
(QPochhammer[x^5, x^5]/QPochhammer[x, x]^5 + O[x]^40)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)

first(n)=my(x='x); Vec(prod(k=1, n, (1-x^(5*k))/(1-x^k)^5, 1+O(x^(n+1)))) \\ Charles R Greathouse IV, Nov 07 2016

x='x+O('x^66); Vec(eta(x^5)/eta(x)^5) \\ Joerg Arndt, Nov 27 2016

G.f.: Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^5.
a(n) ~ exp(4*Pi*sqrt(n/5)) / (sqrt(2) * 5^(7/4) * n^(7/4)). - Vaclav Kotesovec, Nov 10 2016
G.f.: (x^5; x^5)inf/((x; x)_inf)^5, where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2016
a(0) = 1, a(n) = (5/n)*Sum_{k=1..n} A285896(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017

A132972 Expansion of chi(q)^3 / chi(q^3) in powers of q where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, 3, 3, 3, 6, 9, 12, 15, 21, 30, 36, 45, 60, 78, 96, 117, 150, 189, 228, 276, 342, 420, 504, 603, 732, 885, 1050, 1245, 1488, 1773, 2088, 2454, 2901, 3420, 3996, 4662, 5460, 6378, 7404, 8583, 9972, 11565, 13344, 15378, 17748, 20448, 23472, 26910, 30876

Offset: 0

Michael Somos, Sep 06 2007

nonn

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

			G.f. = 1 + 3*q + 3*q^2 + 3*q^3 + 6*q^4 + 9*q^5 + 12*q^6 + 15*q^7 + 21*q^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. A062244, A132975, A273845.

nmax = 60; CoefficientList[Series[Product[(1 + x^(2*k-1))^3 / (1 + x^(6*k-3)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2]^3 / QPochhammer[ -q^3, q^6], {q, 0, n}]; (* Michael Somos, Oct 31 2015 *)

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

Expansion of eta(q^2)^6 * eta(q^3) * eta(q^12) / (eta(q)^3 * eta(q^4)3 * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ 3, -3, 2, 0, 3, -2, 3, 0, 2, -3, 3, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (2 + u*v) * (u*v - 1)^3 - (u - u^4) * (v - v^4).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u * (4 - 2*u + u^2) - v^3 * (1 + u + u^2).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (2 + u1 * u2) - u3 * u6 * (1 + u1 + u2).
G.f. is a period 1 Fourier series which satisfies f(-1/(144*t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A062244.
G.f.: Product_{k>0} (1 + x^(2*k-1))^3 / (1 + x^(6*k-3)).
a(n) = 3 * A132975(n) unless n=0.
Empirical: Sum_{n>=1} exp(-Pi)^(n-1)*a(n) = (-2 + 2*sqrt(3))^(1/3). - Simon Plouffe, Feb 20 2011
a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Sep 08 2015
It appears that the g.f. A(x) = F(x)^3, where F(x) = exp( Sum_{n >= 0} x^(3*n+1)/((3*n + 1)*(1 - (-1)^(n+1)*x^(3*n+1))) + x^(3*n+2)/((3*n + 2)*(1 - (-1)^n*x^(3*n + 2))) ). Cf. A273845. - Peter Bala, Dec 23 2021

A277968 Expansion of ((Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^3) - 1)/3 in powers of x.

Original entry on oeis.org

0, 1, 3, 7, 16, 33, 66, 125, 231, 412, 720, 1227, 2056, 3380, 5478, 8745, 13792, 21483, 33114, 50510, 76344, 114356, 169920, 250503, 366666, 532975, 769758, 1104847, 1576640, 2237331, 3158208, 4435502, 6199479, 8624820, 11946096, 16475880, 22630864, 30962990

Offset: 0

Seiichi Manyama, Nov 07 2016

nonn

			G.f. = x + 3*x^2 + 7*x^3 + 16*x^4 + 33*x^5 + 66*x^6 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. Expansion of ((Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k) - 1)/k in powers of x: A014968 (k=2), this sequence (k=3), A277974 (k=5), A160549 (k=7), A277912 (k=11).

nmax = 50; CoefficientList[Series[(Product[(1 - x^(3*j))/(1 - x^j)^3, {j, 1, nmax}] - 1)/3, {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^3] / QPochhammer[ x]^3 - 1) / 3, {x, 0, n}]; (* Michael Somos, Nov 13 2016 *)

first(n)=my(x='x); concat([0], Vec((prod(k=1, n, (1-x^(3*k))/(1-x^k)^3, 1+O(x^(n+1)))-1)/3)) \\ Charles R Greathouse IV, Nov 07 2016

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A) / eta(x + A)^3 - 1) / 3, n))}; /* Michael Somos, Nov 13 2016 */

a(n) = A273845(n)/3, n > 0.
G.f.: ((Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^3) - 1)/3.
a(n) ~ exp(4*Pi*sqrt(n)/3) / (27*sqrt(2)*n^(5/4)). - Vaclav Kotesovec, Nov 10 2016

A274327 Expansion of Product_{n>=1} (1 - x^(4*n))/(1 - x^n)^4 in powers of x.

Original entry on oeis.org

1, 4, 14, 40, 104, 248, 560, 1200, 2474, 4924, 9520, 17928, 33008, 59528, 105408, 183536, 314744, 532208, 888382, 1465208, 2389808, 3857456, 6166096, 9766576, 15336816, 23888844, 36924656, 56659296, 86341664, 130710104, 196640576, 294059872, 437232746, 646561792

Offset: 0

Seiichi Manyama, Nov 07 2016

nonn

			G.f.: 1 + 4*x + 14*x^2 + 40*x^3 + 104*x^4 + 248*x^5 + 560*x^6 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol

Cf. Expansion of Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k in powers of x: A015128 (k=2), A273845 (k=3), this sequence (k=4), A277212 (k=5), A277283 (k=6), A160539 (k=7).

Cf. A083703.

nmax = 50; CoefficientList[Series[Product[(1 - x^(4*k))/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *)
(QPochhammer[x^4, x^4]/QPochhammer[x, x]^4 + O[x]^40)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)

first(n)=my(x='x);Vec(prod(k=1,n,(1-x^(4*k))/(1-x^k)^4,1+O(x^(n+1)))) \\ Charles R Greathouse IV, Nov 07 2016

G.f.: Product_{n>=1} (1 - x^(4*n))/(1 - x^n)^4.
a(n) ~ 5*exp(Pi*sqrt(5*n/2)) / (2^(13/2) * n^(3/2)). - Vaclav Kotesovec, Nov 10 2016
G.f.: (x^4; x^4)inf/((x; x)_inf)^4, where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2016
a(0) = 1, a(n) = (4/n)*Sum_{k=1..n} A285895(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017

A278690 Expansion of Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^2 in powers of x.

Original entry on oeis.org

1, 2, 5, 9, 18, 31, 54, 88, 144, 225, 351, 531, 800, 1179, 1728, 2492, 3573, 5058, 7119, 9918, 13743, 18882, 25810, 35028, 47313, 63513, 84883, 112833, 149373, 196803, 258309, 337590, 439650, 570357, 737496, 950270, 1220688, 1563021, 1995642, 2540466, 3225386

Offset: 0

Seiichi Manyama, Nov 26 2016

			G.f. = 1 + 2*x + 5*x^2 + 9*x^3 + 18*x^4 + 31*x^5 + 54*x^6 + ...
G.f. = q + 2*q^25 + 5*q^49 + 9*q^73 + 18*q^97 + 31*q^121 + 54*q^145 + ... - _Michael Somos_, Nov 25 2019

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^k: A000726 (k=1), this sequence (k=2), A273845 (k=3), A182819 (k=4).
Cf. Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^2: A000041 (k=1), A015128 (k=2), this sequence (k=3), A160461 (k=5).
Cf. A298311.

nmax = 50; CoefficientList[Series[Product[(1 - x^(3*k))/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 26 2016 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3] / QPochhammer[ x]^2, {x, 0, n}]; (* Michael Somos, Nov 25 2019 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) / eta(x + A)^2, n))}; /* Michael Somos, Nov 25 2019 */

G.f.: Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^2.
a(n) ~ sqrt(5/3)*exp(sqrt(10*n)*Pi/3)/(12*n). - Vaclav Kotesovec, Nov 26 2016
Expansion of q^(-1/24) * eta(q^3) / eta(q)^2 in powers of q. - Michael Somos, Nov 25 2019
G.f.: 1/Product_{n > = 1} ( 1 - x^(n/gcd(n,k)) ) for k = 3. Cf. A000041 (k = 1), A015128 (k = 2), A298311 (k = 4) and A160461 (k = 5). - Peter Bala, Nov 17 2020

A277283 Expansion of Product_{n>=1} (1 - x^(6*n))/(1 - x^n)^6 in powers of x.

Original entry on oeis.org

1, 6, 27, 98, 315, 918, 2491, 6366, 15498, 36182, 81501, 177876, 377558, 781626, 1582173, 3137832, 6108051, 11687598, 22012816, 40855674, 74799828, 135210868, 241511115, 426570624, 745516240, 1290006276, 2211202692, 3756468658, 6327617862, 10572763842

Offset: 0

Seiichi Manyama, Nov 07 2016

nonn

			G.f.: 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 918*x^5 + 2491*x^6 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol

Cf. Expansion of Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k in powers of x: A015128 (k=2), A273845 (k=3), A274327 (k=4), A277212 (k=5), this sequence (k=6), A160539 (k=7).

(QPochhammer[x^6, x^6]/QPochhammer[x, x]^6 + O[x]^40)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^(6*k))/(1 - x^k)^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 21 2016 *)

first(n)=my(x='x); Vec(prod(k=1, n, (1-x^(6*k))/(1-x^k)^6, 1+O(x^(n+1)))) \\ Charles R Greathouse IV, Nov 07 2016

G.f.: Product_{n>=1} (1 - x^(6*n))/(1 - x^n)^6.
G.f.: (x^6; x^6)inf/((x; x)_inf)^6, where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2016
a(n) ~ 35*sqrt(35) * exp(sqrt(35*n)*Pi/3) / (3456*sqrt(3)*n^2). - Vaclav Kotesovec, Nov 21 2016

cp's OEIS Frontend

A005882 Theta series of planar hexagonal lattice (A2) with respect to deep hole.

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A277212 Expansion of Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^5 in powers of x.

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A132972 Expansion of chi(q)^3 / chi(q^3) in powers of q where chi() is a Ramanujan theta function.

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A277968 Expansion of ((Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^3) - 1)/3 in powers of x.

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A274327 Expansion of Product_{n>=1} (1 - x^(4*n))/(1 - x^n)^4 in powers of x.

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A278690 Expansion of Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^2 in powers of x.

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A277283 Expansion of Product_{n>=1} (1 - x^(6*n))/(1 - x^n)^6 in powers of x.

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