A000729 -id:A000729 - cp's OEIS frontend

A227695 Expansion of psi(x)^2 * phi(-x)^6 in powers of x where phi(), psi() are Ramanujan theta functions.

1, -10, 37, -50, -30, 128, -25, -34, -320, 310, 410, -370, -87, -410, 320, 30, 500, 384, -630, -640, -359, 300, -326, 2560, -110, -1098, -1280, -370, 1490, -1850, 269, 1500, 1216, 640, 570, -3328, 340, -2010, -1110, 1790, 768, 3200, 303, 750, -1600, -442

Offset: 0

Michael Somos, Sep 02 2013

sign

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

			G.f. = 1 - 10*x + 37*x^2 - 50*x^3 - 30*x^4 + 128*x^5 - 25*x^6 - 34*x^7 - 320*x^8 + ...
G.f. = q - 10*q^5 + 37*q^9 - 50*q^13 - 30*q^17 + 128*q^21 - 25*q^25 - 34*q^29 + ...

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. A000729, A104794, A227317.

a[ n_] := SeriesCoefficient[ (QPochhammer[ x]^5 / QPochhammer[ x^2])^2, {x, 0, n}];

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^5 / eta(x^2 + A))^2, n))};

Expansion of q^(-1/4) * (eta(q)^5 / eta(q^2))^2 in powers of q.
Expansion of phi(-x)^5 * f(-x^2)^3 = phi(-x)^2 * f(-x)^6 in powers of x where phi(), f() are Ramanujan theta functions.
Euler transform of period 2 sequence [ -10, -8, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 8192 (t / i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227317.
G.f.: (Product_{k>0} (1 - x^k)^5 / (1 - x^(2*k)))^2.
Convolution of A000729 and A104794.

A319933 A(n, k) = [x^k] DedekindEta(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, -1, -2, 1, 0, 0, -1, -3, 1, 0, 0, 2, 0, -4, 1, 0, 1, 1, 5, 2, -5, 1, 0, 0, 2, 0, 8, 5, -6, 1, 0, 1, -2, 0, -5, 10, 9, -7, 1, 0, 0, 0, -7, -4, -15, 10, 14, -8, 1, 0, 0, -2, 0, -10, -6, -30, 7, 20, -9, 1, 0, 0, -2, 0, 8, -5, 0, -49, 0, 27, -10, 1

Offset: 0

Peter Luschny, Oct 02 2018

The columns are generated by polynomials whose coefficients constitute the triangle of signed D'Arcais numbers A078521 when multiplied with n!.

			[ 0] 1,   0,   0,    0,     0,    0,     0,     0,     0,     0, ... A000007
[ 1] 1,  -1,  -1,    0,     0,    1,     0,     1,     0,     0, ... A010815
[ 2] 1,  -2,  -1,    2,     1,    2,    -2,     0,    -2,    -2, ... A002107
[ 3] 1,  -3,   0,    5,     0,    0,    -7,     0,     0,     0, ... A010816
[ 4] 1,  -4,   2,    8,    -5,   -4,   -10,     8,     9,     0, ... A000727
[ 5] 1,  -5,   5,   10,   -15,   -6,    -5,    25,    15,   -20, ... A000728
[ 6] 1,  -6,   9,   10,   -30,    0,    11,    42,     0,   -70, ... A000729
[ 7] 1,  -7,  14,    7,   -49,   21,    35,    41,   -49,  -133, ... A000730
[ 8] 1,  -8,  20,    0,   -70,   64,    56,     0,  -125,  -160, ... A000731
[ 9] 1,  -9,  27,  -12,   -90,  135,    54,   -99,  -189,   -85, ... A010817
[10] 1, -10,  35,  -30,  -105,  238,     0,  -260,  -165,   140, ... A010818
    A001489,  v , A167541, v , A319931,  v ,         diagonal: A008705
           A080956       A319930      A319932

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Fifth ed., Clarendon Press, Oxford, 2003.

M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
Steven R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
Vaclav Kotesovec, The integration of q-series
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
M. Newman, A table of the coefficients of the powers of eta(tau), Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216. [Annotated scanned copy]
Tim Silverman, Counting Cliques in Finite Distant Graphs, arXiv preprint arXiv:1612.08085 [math.CO], 2016.
Michael Somos, Introduction to Ramanujan theta functions
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Transpose of A286354.

Cf. A078521, A319574 (JacobiTheta3).

# DedekindEta is defined in A000594
for n in 0:10
    DedekindEta(10, n) |> println
end

DedekindEta := (x, n) -> mul(1-x^j, j=1..n):
A319933row := proc(n, len) series(DedekindEta(x, len)^n, x, len+1):
seq(coeff(%, x, j), j=0..len-1) end:
seq(print([n], A319933row(n, 10)), n=0..10);

eta[x_, n_] := Product[1 - x^j, {j, 1, n}];
A[n_, k_] := SeriesCoefficient[eta[x, k]^n, {x, 0, k}];
Table[A[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 10 2018 *)

from sage.modular.etaproducts import qexp_eta
def A319933row(n, len):
    return (qexp_eta(ZZ['q'], len+4)^n).list()[:len]
for n in (0..10):
    print(A319933row(n, 10))

A209941 Expansion of f(x)^6 in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, 6, 9, -10, -30, 0, 11, -42, 0, 70, 18, 54, 49, -90, 0, 22, -60, 0, -110, 0, 81, -180, -78, 0, 130, 198, 0, 182, -30, -90, 121, -84, 0, 0, 210, 0, -252, 102, -270, -170, 0, 0, -69, -330, 0, 38, 420, 0, -190, 390, 0, 108, 0, 0, 0, 300, 99, -442, 210, 0, 418

Offset: 0

Michael Somos, Mar 16 2012

sign

Number 59 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. = 1 + 6*x + 9*x^2 - 10*x^3 - 30*x^4 + 11*x^6 - 42*x^7 + 70*x^9 + ...
G.f. = q + 6*q^5 + 9*q^9 - 10*q^13 - 30*q^17 + 11*q^25 - 42*q^29 + 70*q^37 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
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. A000729, A133089, A208845.

seq(coeff(series(mul((1-(-x)^k)^6,k=1..n), x,n+1),x,n),n=0..70); # Muniru A Asiru, Aug 12 2018

a[ n_] := SeriesCoefficient[QPochhammer[ x^2]^18 / (QPochhammer[ x] QPochhammer[ x^4])^6, {x, 0, n}]; (* Michael Somos, Jun 09 2015 *)
CoefficientList[Series[(QPochhammer[x^2]^3/(QPochhammer[x] QPochhammer[x^4]))^6, {x, 0, 50}], x] (* G. C. Greubel, Aug 11 2018 *)

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

{a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, A = factor(4*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p%4==3, if( e%2, 0, p^e), for( i=1, sqrtint(p\2), if( issquare(p - i^2, &y), x=i; break)); a0=1; a1 = y = 2 * (x^2 - y^2) * (-1)^((p%8==5) + y); for( i=2, e, x = y*a1 - p^2*a0; a0=a1; a1=x); a1)))};

Expansion of q^(-1/4) * (eta(q^2)^3 / (eta(q) * eta(q^4)))^6 in powers of q.
Euler transform of period 4 sequence [6, -12, 6, -6, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 512 (t/i)^3 f(t) where q = exp(2 Pi i t).
a(n) = b(4*n + 1) where b(n) is multiplicative b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * p^e if p == 3 (mod 4), else b(p^e) = b(p) * b(p^(e-1)) - p^2 * b^(p^(e-2)) if p == 1 (mod 4).
G.f.: Product_{k>0} (1 - (-x)^k)^6.
a(n) = (-1)^n * A000729(n). a(9*n + 5) = a(9*n + 8) = 0. a(9*n + 2) = 9 * a(n).
Convolution square of A133089. Convolution cube of A208845. - Michael Somos, Jun 09 2015

A258739 Expansion of (f(-x)^3 / f(-x^2))^6 - 64 * x * (f(-x^2)^3 / f(-x))^6 in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, -82, -243, -1194, 2242, 0, 3599, 2950, 0, -12242, -20950, 19926, -16807, 7294, 0, 18950, 97908, 0, -88806, 0, 59049, -183844, 51050, 0, -92142, -98002, 0, 246486, 118706, 290142, -161051, -38868, 0, 0, 75658, 0, -241900, 47614, -544806, -493658, 0, 0

Offset: 0

Michael Somos, Jun 08 2015

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This is a member of an infinite family of integer weight modular forms. g_1 = A008441, g_2 = A002171, g_3 = A000729, g_4 = A215601, g_5 = A215472.
Denoted by g_6(q) in Cynk and Hulek on page 8 as a level 32 cusp form of weight 6.

			G.f. = 1 - 82*x - 243*x^2 - 1194*x^3 + 2242*x^4 + 3599*x^6 + 2950*x^7 + ...
G.f. = q - 82*q^5 - 243*q^9 - 1194*q^13 + 2242*q^17 + 3599*q^25 + 2950*q^29 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
S. Cynk and K. Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds (arXiv:math/0509424).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000729, A002171, A008441, A215472, A215601.

A := Basis( CuspForms( Gamma0(32), 6), 165); A[1]  - 82*A[5] - 243*A[9] - 1194*A[13] + 2242*A[16];

a[ n_] := SeriesCoefficient[ (QPochhammer[ x]^3 / QPochhammer[ x^2])^6 - 64 x (QPochhammer[ x^2]^3 / QPochhammer[ x])^6, {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 / eta(x^2 + A))^6 - 64 * x * (eta(x^2 + A)^3 / eta(x + A))^6, n))};

{a(n) = my(A, p, e, x, y, a0, a1); if(n<0, 0, n = 4*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k,]; if(p==2, 0, p%4==3, if(e%2, 0, (-p)^(5*e/2)), y = -sum(i=0, p-1, kronecker(i^3-i, p)); a0=2; a1=y; for(i=2, 5, x=y*a1 -p*a0; a0=a1; a1=x); y=a1; a0=1; a1=y; for(i=2, e, x=y*a1 -p^5*a0; a0=a1; a1=x); a1)))};

Expansion of q^(-1/4) * ((eta(-q)^3 / eta(-q^2))^6 - 64 * (eta(-q^2) / eta(-q))^6) in powers of q.
a(n) = b(4*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^(5*e/2) if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - p^4 * b(p^(e-2)) if p == 1 (mod 4).
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = -(32^3) (t/i)^6 f(t) where q = exp(2 Pi i t).

cp's OEIS Frontend

A227695 Expansion of psi(x)^2 * phi(-x)^6 in powers of x where phi(), psi() are Ramanujan theta functions.

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A319933 A(n, k) = [x^k] DedekindEta(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.

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A209941 Expansion of f(x)^6 in powers of x where f() is a Ramanujan theta function.

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A258739 Expansion of (f(-x)^3 / f(-x^2))^6 - 64 * x * (f(-x^2)^3 / f(-x))^6 in powers of x where f() is a Ramanujan theta function.

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