A187150 -id:A187150 - cp's OEIS frontend

A000727 Expansion of Product_{k >= 1} (1 - x^k)^4.

1, -4, 2, 8, -5, -4, -10, 8, 9, 0, 14, -16, -10, -4, 0, -8, 14, 20, 2, 0, -11, 20, -32, -16, 0, -4, 14, 8, -9, 20, 26, 0, 2, -28, 0, -16, 16, -28, -22, 0, 14, 16, 0, 40, 0, -28, 26, 32, -17, 0, -32, -16, -22, 0, -10, 32, -34, -8, 14, 0, 45, -4, 38, 8, 0, 0, -34, -8, 38, 0, -22, -56, 2, -28, 0, 0, -10, 20, 64, -40, -20, 44

Offset: 0

N. J. A. Sloane

Number 51 of the 74 eta-quotients listed in Table I of Martin (1996).

Ramanujan (see the link, pp. 155 and 157 Nr. 23.) conjectured the expansion coefficients called Psi_4(n) of eta^4(6*z) in powers of q = exp(2*Pi*i*z), Im(z) > 0, where i is the imaginary unit. In the Finch link on p. 5, multiplicity is used and Psi_4(p^r), called f(p^r), is given (see also b(p^e) formula given by Michael Somos, Aug 23 2006). Mordell proved this conjecture on pp. 121-122 based on Klein-Fricke, Theorie der elliptischen Modulfunktionen, 1892, Band II, p. 374. The product formula for the Dirichlet series, Mordell, eq. (7) for a=2,is used to find Psi_4(n), called f_2(n), from f_2(p) for primes p. The primes p = 2 and 3 do not appear in the product. - Wolfdieter Lang, May 03 2016

			G.f. = 1 - 4*x + 2*x^2 + 8*x^3 - 5*x^4 - 4*x^5 - 10*x^6 + 8*x^7 + 9*x^8 + ...
G.f. = q - 4*q^7 + 2*q^13 + 8*q^19 - 5*q^25 - 4*q^31 - 10*q^37 + 8*q^43 + ...

Morris 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.
J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, p. 415. Exer. 47.2.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389.
Amanda Clemm, Modular Forms and Weierstrass Mock Modular Forms, Mathematics, volume 4, issue 1, (2016).
S. Cooper, M. D. Hirschhorn and R. Lewis, Powers of Euler's Product and Related Identities, The Ramanujan Journal, Vol. 4 (2), 137-155 (2000).
S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Louis J. Mordell, On Mr. Ramanujan's empirical expansions of modular functions, Proceedings of the Cambridge Philosophical Society 19 (1917), pp. 117-124.
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]
S. Ramanujan, On certain Arithmetical Functions, Trans. Cambridge Philos. Soc. 22 (1916) 159-184; also in Collected papers, eds. G. H. Hardy, P. V. Seshu Aiyar and B. M. Wilson, Chelsea publ. Comp., 1962 (reprint from CUP 1927), pp. 136-162, 340- 341.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Robert M. Ziff, On Cardy's formula for the critical crossing probability in 2d percolation, J. Phys. A. 28, 1249-1255 (1995).
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Cf. A000731, A002107, A023003, A187076, A187150, A258404, A280328.

Powers of Euler's product: A000594, A000727 - A000731, A000735, A000739, A002107, A010815 - A010840.

# DedekindEta is defined in A000594.
L000727List(len) = DedekindEta(len, 4)
L000727List(82) |> println # Peter Luschny, Mar 09 2018

qEigenform( EllipticCurve( [0, 0, 0, 0, 1]), 493); /* Michael Somos, Jun 12 2014 */

A := Basis( ModularForms( Gamma0(36), 2), 493); A[2] - 4*A[8]; /* Michael Somos, Jun 12 2014 */

Basis( CuspForms( Gamma0(36), 2), 493)[1]; /* Michael Somos, May 17 2015 */

Coefficients(&*[(1-x^m)^4:m in [1..100]])[1..100] where x is PolynomialRing(Integers()).1; // Vincenzo Librandi, Mar 10 2018

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> -4): seq(a(n), n=0..81); # Alois P. Heinz, Sep 08 2008

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b]; a = etr[-4&]; Table[a[n], {n, 0, 81}] (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^4, {x, 0, n}]; (* Michael Somos, Jun 12 2014 *)
nmax = 80; CoefficientList[Series[Sum[Sum[(-1)^(k+m) * (2*k+1) * q^(k*(k+1)/2 + m*(3*m-1)/2), {k, 0, nmax}], {m, -nmax, nmax}], {q, 0, nmax}], q] (* Vaclav Kotesovec, Dec 06 2015 *)

{a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 6*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p<5, 0, p%6==5, if(e%2, 0, (-1)^(e/2) * p^(e/2)), for( y=1, sqrtint(p\3), if( issquare( p - 3*y^2, &x), break)); a0=1; if( x%3!=1, x=-x); a1 = x = 2*x; for( i=2, e, y = x*a1 - p*a0; a0=a1; a1=y); a1)))}; /* Michael Somos, Aug 23 2006 */

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

{a(n) = if( n<0, 0, ellak( ellinit( [0, 0, 0, 0, 1], 1), 6*n + 1))}; /* Michael Somos, Jul 01 2004 */

ModularForms( Gamma0(36), 2, prec=493).0; # Michael Somos, Jun 12 2014

Euler transform of period 1 sequence [-4, -4, ...]. - Michael Somos, Apr 02 2005
Given g.f. A(x), then B(q) = q * A(q^3)^2 satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = w*u^2 - v^3 + 16 * u*w^2. - Michael Somos, Apr 02 2005
a(n) = b(6*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)), b(p) = 0 if p == 5 (mod 6), b(p) = 2*x where p = x^2 + 3*y^2 == 1 (mod 6) and x == 1 (mod 3). - Michael Somos, Aug 23 2006
Coefficients of L-series for elliptic curve "36a1": y^2 = x^3 + 1. - Michael Somos, Jul 01 2004
a(n) = (-1)^n * A187076(n). a(2*n + 1) = -4 * A187150(n). a(25*n + 9) = a(25*n + 14) = a(25*n + 19) = a(25*n + 24) = 0. a(25*n + 4) = -5 * a(n). Convolution inverse of A023003. Convolution square of A002107. Convolution square is A000731.
a(0) = 1, a(n) = -(4/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 26 2017
G.f.: exp(-4*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018
Let M = p_1*...*p_k be a positive integer whose prime factors p_i (not necessarily distinct) are all congruent to 5 (mod 6). Then a( M^2*n + (M^2 - 1)/6 ) = (-1)^k*M*a(n). See Cooper et al., equation 4. - Peter Bala, Dec 01 2020
a(n) = b(6*n + 1) where b() is multiplicative with b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * (-p)^(e/2) if p == 2 (mod 3), b(p^e) = (((x+sqrt(-3)*y)/2)^(e+1) - ((x-sqrt(-3)*y)/2)^(e+1))/x if p == 1 (mod 3) where p = x^2 + 3*y^2 and x == 1 (mod 3). - Jianing Song, Mar 19 2022

A187076 Coefficients of L-series for elliptic curve "144a1": y^2 = x^3 - 1.

Original entry on oeis.org

1, 4, 2, -8, -5, 4, -10, -8, 9, 0, 14, 16, -10, 4, 0, 8, 14, -20, 2, 0, -11, -20, -32, 16, 0, 4, 14, -8, -9, -20, 26, 0, 2, 28, 0, 16, 16, 28, -22, 0, 14, -16, 0, -40, 0, 28, 26, -32, -17, 0, -32, 16, -22, 0, -10, -32, -34, 8, 14, 0, 45, 4, 38, -8, 0, 0, -34

Offset: 0

Michael Somos, Mar 05 2011

sign

Number 67 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 + 4*x + 2*x^2 - 8*x^3 - 5*x^4 + 4*x^5 - 10*x^6 - 8*x^7 + 9*x^8 + ...
G.f. = q + 4*q^7 + 2*q^13 - 8*q^19 - 5*q^25 + 4*q^31 - 10*q^37 - 8*q^43 + 9*q^49 + ...

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Amanda Clemm, Modular Forms and Weierstrass Mock Modular Forms, Mathematics, volume 4, issue 1, (2016)
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. A000727, A187150, A209676, A258779, A280384.

A := Basis( CuspForms( Gamma0(144), 2), 398); A[1] + 4*A[7] + 2*A[11] - 8*A[13]; /* Michael Somos, Jan 01 2017 */

a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^4, {x, 0, n}]; (* Michael Somos, Jun 10 2015 *)

{a(n) = if( n<0, 0, ellak( ellinit( [0, 0, 0, 0, -1], 1), 6*n + 1))};

{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)))^4, n))};

{a(n) = my(m, A, p, e, x, y, a0, a1); if( n<0, 0, m = 6*n + 1; A = factor(m); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p<5, 0, p%6==5, if(e%2, 0, (-p)^(e/2)), for( y=1, sqrtint(p\3), if( issquare(p - 3*y^2, &x), break)); a0 = 1; if( x%6>3, x = -x); a1 = x = 2*x; for( i=2, e, y = x*a1 - p*a0; a0 = a1; a1 = y); a1)))};

Expansion of f(x)^4 in powers of x where f() is a Ramanujan theta function.
Expansion of q^(-1/6) * (eta(q^2)^3 / (eta(q) * eta(q^4)))^4 in powers of q.
Euler transform of period 4 sequence [4, -8, 4, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = f(t) where q = exp(2 Pi i t).
a(n) = b(6*n + 1) and b(n) is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)), b(p) = 0 if p == 5 (mod 6), b(p) = 2 * x where p = x^2 + 3 * y^2 == 1 (mod 6) and x == 4, 5 (mod 6).
G.f.: Product_{k>0} (1 - (-x)^k)^4. a(n) = (-1)^n * A000727(n).
Convolution cube is A209676. - Michael Somos, Jun 10 2015
a(2*n) = A258779(n). a(2*n + 1) = 4 * A187150(n). - Michael Somos, Jun 10 2015

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A000727 Expansion of Product_{k >= 1} (1 - x^k)^4.

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A187076 Coefficients of L-series for elliptic curve "144a1": y^2 = x^3 - 1.

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