A030211 -id:A030211 - cp's OEIS frontend

A002288 G.f.: q * Product_{m>=1} (1-q^m)^8*(1-q^2m)^8.

0, 1, -8, 12, 64, -210, -96, 1016, -512, -2043, 1680, 1092, 768, 1382, -8128, -2520, 4096, 14706, 16344, -39940, -13440, 12192, -8736, 68712, -6144, -34025, -11056, -50760, 65024, -102570, 20160, 227552, -32768, 13104, -117648, -213360, -130752, 160526, 319520

Offset: 0

N. J. A. Sloane

This is Glaisher's Theta(n). - N. J. A. Sloane, Nov 26 2018
Number 2 of the 74 eta-quotients listed in Table I of Martin (1996).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = q - 8*q^2 + 12*q^3 + 64*q^4 - 210*q^5 - 96*q^6 + 1016*q^7 - 512*q^8 + ...

J. W. L. Glaisher, On the representation of a number as a sum of 14 and 16 squares, Quart. J. Math. 38 (1907), 178-236 (see p. 198).
F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg 1994 p 133.
G. Shimura, Modular forms of half-integral weight, pp. 57-74 of Modular Functions of One Variable I (Antwerp 1972), Lect. Notes Math. 320 (1973).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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 (first 1002 terms from T. D. Noe)
T. Ishikawa, Congruences between binomial coefficients binomial(2f,f) and Fourier coefficients of certain eta-products, Hiroshima Math. J. 22 (1992), no. 3, 583-590.
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
LMFDB, Newform orbit 2.8.a.a.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
H Movasati and Y Nikdelan, Gauss-Manin Connection in Disguise: Dwork Family, arXiv preprint arXiv:1603.09411 [math.AG], 2016-2021.
H.-G. Quebbemann, Lattices with theta-functions for G(sqrt(2)) and linear codes, J. Algebra, 105 (1987), 443-450.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Index entries for sequences mentioned by Glaisher

Cf. A030211.

Basis( CuspForms( Gamma0(2), 8), 100) [1]; /* Michael Somos, Dec 09 2013 */

t1 := product((1-q^m)^8,m=1..40): subs(q=q^2,t1): series(q*t1*%,q,40);

max = 36; f[q_] := q*Product[(1-q^m)^8*(1-q^(2m))^8, {m, 1, max}]; CoefficientList[ Series[f[q], {q, 0, max}], q] (* Jean-François Alcover, Jul 18 2012 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^2])^8, {q, 0, n}]; (* Michael Somos, Apr 09 2013 *)
a[ n_] := SeriesCoefficient[(EllipticTheta[ 4, 0, q] EllipticTheta[ 2, 0, q^(1/2)] / 2)^8, {q, 0, n}]; (* Michael Somos, Dec 09 2013 *)

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^2 + A))^8, n))}; /* Michael Somos, Jul 16 2004 */

q='q+O('q^50); concat(0, Vec((eta(q)*eta(q^2))^8)) \\ Altug Alkan, Sep 19 2018

CuspForms( Gamma0(2), 8, prec=100).0; # Michael Somos, May 28 2013

Expansion of cusp form (e(1)-e(2))(e(1)-e(3))(e(2)-e(3))^2 for GAMMA_0(2).
Expansion of q * psi(q)^8 * phi(-q)^8 in powers of q where psi(), phi() are Ramanujan theta functions. - Michael Somos, Dec 09 2013
Expansion of (eta(q) * eta(q^2))^8 in powers of q. - Michael Somos, Mar 18 2003
Euler transform of period 2 sequence [ -8, -16, ... ].
a(n) is multiplicative with a(2^e) = (-8)^e, a(p^e) = a(p) * a(p^(e-1)) - p^7 * a(p^(e-2)). - Michael Somos, Mar 08 2006
Given A = A0 + A1 + A2 + A3 is the 4-section, then 0 = A2^3 + 2 * A0 * (A1^2 + A3^2) - 4 * A1*A2*A3 - 3 * A0^2*A2. - Michael Somos, Mar 08 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 16 (t/i)^8 f(t) where q = exp(2 Pi i t). - Michael Somos, Apr 09 2013
a(2*n) = -8 * a(n). Convolution square of A030211. - Michael Somos, Apr 09 2013
G.f.: x*exp(8*Sum_{k>=1} (sigma(2*k) - 4*sigma(k))*x^k/k). - Ilya Gutkovskiy, Sep 19 2018

Extended, and better description added by N. J. A. Sloane, Jan 15 1996

A134461 Expansion of (phi(x) * psi(-x))^4 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 4, -2, -24, -11, 44, 22, -8, 50, -44, -96, 56, -121, -152, 198, 160, 176, 48, -162, 88, -198, -52, 22, -528, 233, 200, -242, -88, -176, 668, 550, 264, -44, -188, 224, -728, 154, -484, -1056, 656, -311, -236, -100, 792, 714, -528, 640, 88, -478, -484, 1566, 968, 192, 780, -1994, -648, -942

Offset: 0

Michael Somos, Oct 26 2007

sign

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

Number 34 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 + 4*x - 2*x^2 - 24*x^3 - 11*x^4 + 44*x^5 + 22*x^6 - 8*x^7 + ...
G.f. = q + 4*q^3 - 2*q^5 - 24*q^7 - 11*q^9 + 44*q^11 + 22*q^13 - 8*q^15 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A216711.

The same as A030211 except for signs.

A := Basis( CuspForms( Gamma0(16), 4), 115); A[1] + 4*A[3]; /* Michael Somos, Jun 10 2015 */

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^2] EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8)))^4, {x, 0, n}]; (* Michael Somos, Jun 10 2015 *)

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

Expansion of q^(-1/2) * (eta(q^2)^4 / (eta(q) * eta(q^4)))^4 in powers of q.
Euler transform of period 4 sequence [ 4, -12, 4, -8, ...].
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = b(p)*b(p^(e-1)) - p^3*b(p^(e-2)).
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 256 (t/i)^4 f(t) where q = exp(2 Pi i t).
G.f.: (Product_{k>0} (1 + x^k) * (1 - x^(2*k))^2 / (1 + x^(2*k)))^4.
a(n) = (-1)^n * A030211(n).
Convolution square is A216711. - Michael Somos, Jun 10 2015

A259491 Expansion of (eta(q)^2 * eta(q^2) * eta(q^4)^3 / eta(q^8)^2)^2 in powers of q.

Original entry on oeis.org

1, -4, 0, 16, -16, 8, 0, -96, 112, 44, 0, 176, -448, -88, 0, -32, 1136, -200, 0, -176, -2016, 384, 0, 224, 3136, 484, 0, -608, -5504, -792, 0, 640, 9328, -704, 0, 192, -12112, 648, 0, 352, 14112, 792, 0, -208, -21312, -88, 0, -2112, 31808, -932, 0, 800

Offset: 0

Michael Somos, Jun 28 2015

sign

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

			G.f. = 1 - 4*q + 16*q^3 - 16*q^4 + 8*q^5 - 96*q^7 + 112*q^8 + 44*q^9 + ...

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. A030211, A035016, A131999.

A := Basis( ModularForms( Gamma1(8), 4), 52); A[1] - 4*A[2] + 16*A[4] - 16*A[5] + 8*A[6] - 96*A[8] + 112*A[9];

a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^2 QPochhammer[ q^2] QPochhammer[ q^4]^3 / QPochhammer[ q^8]^2)^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 4, 0, q]^2)^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^2] EllipticTheta[ 4, 0, q^4]^2)^2, {q, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^2 + A) * eta(x^4 + A)^3 / eta(x^8 + A)^2)^2, n))};

Expansion of (phi(q) * phi(q^2) * phi(-q)^2)^2 in powers of q where phi() is a Ramanujan theta function.
Euler transform of period 8 sequence [ -4, -6, -4, -12, -4, -6, -4, -8, ...].
G.f.: Product_{k>0} ((1 - x^k)^4 * (1 + x^k)^2 * (1 + x^(2*k)) / (1 + x^(4*k))^2)^2.
a(2*n + 1) = -4 * A030211(n). a(4*n) = A035016(n). a(4*n + 2) = 0.
Convolution square of A131999.

A289277 a(n) = A005259(n) mod 2*n+1.

Original entry on oeis.org

0, 2, 3, 3, 7, 0, 9, 5, 16, 6, 1, 13, 4, 26, 24, 26, 22, 30, 23, 32, 7, 9, 43, 11, 37, 29, 23, 0, 49, 40, 1, 44, 20, 54, 19, 18, 8, 20, 22, 55, 4, 70, 80, 62, 2, 31, 37, 20, 7, 44, 51, 62, 64, 76, 77, 41, 75, 75, 115, 68, 0, 35, 42, 11, 88, 59, 101, 35, 119, 11

Offset: 0

Seiichi Manyama, Jul 01 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
F. Beukers, Another congruence for the Apéry numbers, J. Number Theory 25 (1987), no. 2, 201-210.
Eric Weisstein's World of Mathematics, Apéry Number

Cf. A005259, A030211, A289278.

Table[Mod[Sum[(Binomial[n, k] Binomial[n + k, k])^2, {k, 0, n}], 2n + 1], {n, 0, 100}] (* Indranil Ghosh, Jul 01 2017 *)

a(n) = sum(k=0, n, (binomial(n, k)*binomial(n+k, k))^2) % (2*n+1); \\ Michel Marcus, Jul 01 2017

If m = 2*n + 1 is a prime, a(n) = A030211(n) mod m.

A289278 a(n) = A005259(n) mod (2*n+1)^2.

Original entry on oeis.org

0, 5, 23, 24, 34, 77, 22, 140, 50, 44, 169, 473, 354, 539, 198, 801, 385, 135, 1207, 617, 1483, 52, 2023, 528, 723, 2273, 2567, 1265, 1303, 2813, 550, 233, 1775, 188, 2365, 728, 154, 1520, 4180, 5585, 571, 236, 3650, 2672, 714, 4581, 4966, 2490, 8931, 4796, 1566

Offset: 0

Seiichi Manyama, Jul 01 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Apéry Number

Cf. A005259, A030211, A289277.

Table[Mod[Sum[(Binomial[n, k] Binomial[n + k, k])^2, {k, 0, n}], (2n + 1)^2], {n, 0, 100}] (* Indranil Ghosh, Jul 01 2017 *)

a(n) = sum(k=0, n, (binomial(n, k)*binomial(n+k, k))^2) % (2*n+1)^2; \\ Michel Marcus, Jul 01 2017

If m = 2*n + 1 is a prime, a(n) = A030211(n) mod m^2.

A291124 Expansion of phi(x)^6 * phi(-x)^2 in powers of x where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 8, 16, -32, -144, -16, 448, 192, -912, -88, 2016, -352, -4032, 176, 5504, 64, -7056, 400, 12112, 352, -18144, -768, 21312, -448, -25536, -968, 35168, 1216, -49536, 1584, 56448, -1280, -56208, 1408, 78624, -384, -109008, -1296, 109760, -704, -114912, -1584

Offset: 0

Michael Somos, Aug 17 2017

sign

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

			G.f. = 1 + 8*x + 16*x^2 - 32*x^3 - 144*x^4 - 16*x^5 + 448*x^6 + 192*x^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. A030211, A045820, A045823, A207541.

A := Basis( ModularForms( Gamma0(16), 4), 42); A[1] + 8*A[2] + 16*A[3] - 32*A[4] - 144*A[5] - 16*A[6] + 448*A[7] + 192*A[8] - 912*A[9];

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x]^6 EllipticTheta[ 4, 0, x]^2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[x^2]^7 / (QPochhammer[ x]^2 QPochhammer[ x^4]^3))^4, {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^7 / (eta(x + A)^2 * eta(x^4 + A)^3))^4, n))};

lista(nn) = {q='q+O('q^nn); Vec((eta(q^2)^7/(eta(q)^2*eta(q^4)^3))^4)} \\ Altug Alkan, Mar 21 2018

Expansion of (eta(q^2)^7 / (eta(q)^2 * eta(q^4)^3))^4 in powers of q.
Euler transform of period 4 sequence [8, -20, 8, -8, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 512 (t/i)^4 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A045820.
G.f.: Product_{k>0} (1 - x^(2*k))^28 / ((1 - x^k)^8 * (1 - x^(4*k))^12).
a(2*n + 1) = 8 * A030211(n). a(4*n + 2) = 16 * A045823(n).
a(2*n) = 16 * (-1)^n * (-sigma_3(n) + sigma_3(n/4)) where sigma_3(n) is the sum of the cubes of the divisors of n if n is an integer else 0.
Convolution square of A207541.

A319456 a(n) = [x^n] Product_{k>=1} ((1 - x^k)(1 - x^(2k)))^n.

Original entry on oeis.org

1, -1, -3, 14, -11, -81, 282, -57, -2043, 5405, 2417, -46476, 94522, 110512, -943407, 1505289, 2807589, -16888311, 23645199, 46006542, -265972791, 472882620, 187884672, -3981273597, 14234579226, -19187383356, -78662039004, 502118911904, -847583768679, -2627514175002

Offset: 0

Ilya Gutkovskiy, Sep 19 2018

sign

Cf. A002171, A002288, A008705, A030204, A030211, A066535, A296043, A296044, A319457.

Table[SeriesCoefficient[Product[((1 - x^k) (1 - x^(2 k)))^n , {k, 1, n}], {x, 0, n}], {n, 0, 29}]
Table[SeriesCoefficient[(QPochhammer[x] QPochhammer[x^2])^n, {x, 0, n}], {n, 0, 29}]
Table[SeriesCoefficient[Exp[n Sum[(DivisorSigma[1, 2 k] - 4 DivisorSigma[1, k]) x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 29}]

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

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

cp's OEIS Frontend

A002288 G.f.: q * Product_{m>=1} (1-q^m)^8*(1-q^2m)^8.

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A134461 Expansion of (phi(x) * psi(-x))^4 in powers of x where phi(), psi() are Ramanujan theta functions.

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A259491 Expansion of (eta(q)^2 * eta(q^2) * eta(q^4)^3 / eta(q^8)^2)^2 in powers of q.

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A289277 a(n) = A005259(n) mod 2*n+1.

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A289278 a(n) = A005259(n) mod (2*n+1)^2.

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A291124 Expansion of phi(x)^6 * phi(-x)^2 in powers of x where phi() is a Ramanujan theta function.

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A319456 a(n) = [x^n] Product_{k>=1} ((1 - x^k)*(1 - x^(2*k)))^n.

A319456 a(n) = [x^n] Product_{k>=1} ((1 - x^k)(1 - x^(2k)))^n.