A000739 -id:A000739 - 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

A002107 Expansion of Product_{k>=1} (1 - x^k)^2.

Original entry on oeis.org

1, -2, -1, 2, 1, 2, -2, 0, -2, -2, 1, 0, 0, 2, 3, -2, 2, 0, 0, -2, -2, 0, 0, -2, -1, 0, 2, 2, -2, 2, 1, 2, 0, 2, -2, -2, 2, 0, -2, 0, -4, 0, 0, 0, 1, -2, 0, 0, 2, 0, 2, 2, 1, -2, 0, 2, 2, 0, 0, -2, 0, -2, 0, -2, 2, 0, -4, 0, 0, -2, -1, 2, 0, 2, 0, 0, 0, -2

Offset: 0

N. J. A. Sloane

Number of partitions of n into an even number of distinct parts minus number of partitions of n into an odd number of distinct parts, with 2 types of each part. E.g., for n=4, we consider k and k* to be different versions of k and so we have 4, 4*, 31, 31*, 3*1, 3*1*, 22*, 211*, 2*11*. The even partitions number 5 and the odd partitions number 4, so a(4)=5-4=1. - Jon Perry, Apr 04 2004
Also, number of partitions of n into parts of -2 different kinds (based upon formal analogy). - Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 29 2004
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 68 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 - 2*x - x^2 + 2*x^3 + x^4 + 2*x^5 - 2*x^6 - 2*x^8 - 2*x^9 + x^10 + ...
G.f. = q - 2*q^13 - q^25 + 2*q^37 + q^49 + 2*q^61 - 2*q^73 - 2*q^97 - 2*q^109 + ...

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 1001 terms from T. D. Noe)
G. E. Andrews, Advanced problems 6562, Amer. Math. Monthly 94, 1987.
M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
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.
J. W. L. Glaisher, On the square of Euler's series, Proc. London Math. Soc., 21 (1889), 182-194.
J. T. Joichi, Hecke-Rogers, Andrews identities; combinatorial proofs, Discrete Mathematics, Vol. 84, Issue 3, 1990, pp. 255-259.
Victor G. Kac and Dale H. Peterson, Infinite-Dimensional Lie Algebras, Theta Functions and Modular Forms, Advances in Mathematics (1984), 53. 125-264, see page 261, (5.19).
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
V. 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.
Tim Silverman, Counting Cliques in Finite Distant Graphs, arXiv preprint arXiv:1612.08085 [math.CO], 2016.
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
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Cf. A000712 (reciprocal of g.f.), A010815, A010816, A258406.

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

# DedekindEta is defined in A000594.
A002107List(len) = DedekindEta(len, 2)
A002107List(78) |> println # Peter Luschny, Mar 09 2018

Basis( CuspForms( Gamma1(144), 1), 926) [1]; /* Michael Somos, May 17 2015 */

A010816 := proc (n); if frac(sqrt(8*n+1)) = 0 then (-1)^((1/2)*isqrt(8*n+1)-1/2)*isqrt(8*n+1) else 0 end if; end proc:
N := 10:
a := proc (n) option remember; if n < 0 then 0 else A010816(n) + add( (-1)^(k+1)*a(n - (1/2)*k*(3*k-1) ), k = -N..-1) + add( (-1)^(k+1)*a(n - (1/2)*k*(3*k-1) ), k = 1..N) end if; end proc:
seq(a(n), n = 0..100); # Peter Bala, Apr 06 2022

terms = 78; Clear[s]; s[n_] := s[n] = Product[(1 - x^k)^2, {k, 1, n}] // Expand // CoefficientList[#, x]& // Take[#, terms]&; s[n = 10]; s[n = 2*n]; While[s[n] != s[n - 1], n = 2*n]; A002107 = s[n] (* Jean-François Alcover, Jan 17 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2, {x, 0, n}]; (* Michael Somos, Jan 31 2015 *)
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, n}]^2, {x, 0, n}]; (* Michael Somos, Jan 31 2015 *)

{a(n) = my(A, p, e, x); if( n<0, 0, n = 12*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p<5, 0, p%12>1, if( e%2, 0, (-1)^((p%12==5) * e/2)), for( i=1, sqrtint(p\9), if( issquare(p - 9*i^2), x=i; break)); (e + 1) * (-1)^(e*x))))}; /* Michael Somos, Aug 30 2006 */

{a(n) = if( n<0, 0, polcoeff( eta(x + x * O(x^n))^2, n))}; /* Michael Somos, Aug 30 2006 */

Vec(eta(x)^2) \\ Charles R Greathouse IV, Apr 22 2016

Expansion of q^(-1/12) * eta(q)^2 in powers of q. - Michael Somos, Mar 06 2012
Euler transform of period 1 sequence [ -2, ...]. - Michael Somos, Mar 06 2012
a(n) = b(12*n + 1) where b(n) is multiplicative and b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 7, 11 (mod 12), b(p^e) = (-1)^(e/2) * (1 + (-1)^e) / 2 if p == 5 (mod 12), b(p^e) = (e + 1) * (-1)^(e*x) if p == 1 (mod 12) where p = x^2 + 9*y^2. - Michael Somos, Sep 16 2006
Convolution inverse of A000712.
a(0) = 1, a(n) = -(2/n)*Sum{k = 0..n-1} a(k)*sigma_1(n-k). - Joerg Arndt, Feb 05 2011
Expansion of f(-x)^2 in powers of x where f() is a Ramanujan theta function. - Michael Somos, May 17 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 12 (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, May 17 2015
a(n) = Sum_{k=0..n} A010815(k)*A010815(n-k); self convolution of A010815. - Gevorg Hmayakyan, Sep 18 2016
G.f.: Sum_{m, n in Z, n >= 2*|m|} (-1)^n * x^((3*(2*n + 1)^2 - (6*m + 1)^2)/24). - Seiichi Manyama, Oct 01 2016
G.f.: exp(-2*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018
From Peter Bala, Jan 02 2021: (Start)
For prime p congruent to 5, 7 or 11 (mod 12), a(n*p^2 + (p^2 - 1)/12) = e*a(n), where e = 1 if p == 7 or 11 (mod 12) and e = -1 if p == 5 (mod 12).
If n and p are coprime then a(n*p + (p^2 - 1)/12) = 0. See Cooper et al., Theorem 1. (End)
With the convention that a(n) = 0 for n < 0 we have the recurrence a(n) = A010816(n) + Sum_{k a nonzero integer} (-1)^(k+1)*a(n - k*(3*k-1)/2), where A010816(n) = (-1)^m*(2*m+1) if n = m*(m + 1)/2, with m positive, is a triangular number else equals 0. For example, n = 10 = (4*5)/2 is a triangular number, A010816(10) = 9, and so a(10) = 9 + a(9) + a(8) - a(5) - a(3) = 9 - 2 - 2 - 2 - 2 = 1. - Peter Bala, Apr 06 2022

A000731 Expansion of Product (1 - x^k)^8 in powers of x.

Original entry on oeis.org

1, -8, 20, 0, -70, 64, 56, 0, -125, -160, 308, 0, 110, 0, -520, 0, 57, 560, 0, 0, 182, -512, -880, 0, 1190, -448, 884, 0, 0, 0, -1400, 0, -1330, 1000, 1820, 0, -646, 1280, 0, 0, -1331, -2464, 380, 0, 1120, 0, 2576, 0, 0, -880, 1748, 0, -3850, 0, -3400, 0, 2703, 4160, -2500, 0, 3458

Offset: 0

N. J. A. Sloane

Number 22 of the 74 eta-quotients listed in Table I of Martin (1996).
Denoted by g_4(q) in Cynk and Hulek in Remark 3.4 on page 12 as the unique level 9 form of weight 4.
This is a member of an infinite family of integer weight modular forms. g_1 = A033687, g_2 = A030206, g_3 = A130539, g_4 = A000731. - Michael Somos, Aug 24 2012
a(n)=0 if and only if A033687(n)=0 (see the Han-Ono paper). - Emeric Deutsch, May 16 2008
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 - 8*x + 20*x^2 - 70*x^3 + 64*x^4 + 56*x^5 - 125*x^6 - 160*x^7 + ...
G.f. = q - 8*q^4 + 20*q^7 - 70*q^13 + 64*q^16 + 56*q^19 - 125*q^25 - ...

Newman, Morris; A table of the coefficients of the powers of eta(tau). Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.
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
Matthew Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
Shaun Cooper, Minchael D. Hirschhorn and Richard Lewis, Powers of Euler's Product and Related Identities, The Ramanujan Journal, Vol. 4 (2), 137-155 (2000).
Slawomir Cynk and Klaus Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds, arXiv:math/0509424 [math.AG], 2005-2006.
Steven R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
Guo-Niu Han and Ken Ono, Hook Lengths and 3-Cores, Ann. Comb. (2011) Vol. 15, 305-312. See also arXiv:0805.2461, [math.NT], 2008.
Iva Kodrnja and Helena Koncul, Polynomials vanishing on a basis of S_m(Gamma_0(N)), Glasnik Matematički (2024) Vol. 59, No. 79, 313-325. See p. 319.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
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. [Annotated scanned copy]
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Cf. A033687, A033690, A092342, A153728.

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

Basis( CuspForms( Gamma0(9), 4), 56) [1]; /* Michael Somos, Dec 09 2013 */

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^8, {x, 0, n}]; (* Michael Somos, Sep 29 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, n}]^8, {x, 0, n}]; (* Michael Somos, Dec 09 2013 *)

{a(n) = if( n<0, 0, polcoeff( eta(x + x * O(x^n))^8, n))};

{a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 3*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 0, p%3==2, if( e%2, 0, (-1)^(e/2) * p^(3*e/2)), forstep( y=sqrtint(4*p\3), sqrtint(p\3), -1, if( issquare( 4*p - 3*y^2, &x), if( x%3!=2, x=-x); break)); a0=1; a1 = y = x * (x^2 - 3*p); for( i=2, e, x = y*a1 - p^3*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Aug 23 2006 */

CuspForms( Gamma0(9), 4, prec=56).0; # Michael Somos, May 28 2013

Expansion of q^(-1/3) * eta(q)^8 in powers of q.
Expansion of q^(-1/3) * b(q)^3 * c(q) / 3 in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Nov 08 2006
Expansion of q^(-1) * b(q) * c(q)^3 / 27 in powers of q^3 where b(), c() are cubic AGM theta functions. - Michael Somos, Nov 08 2006
Euler transform of period 1 sequence [ -8, ...].
a(n) = b(3*n + 1) where b(n) is multiplicative and b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * (-1)^(e/2) * p^(3*e/2) if p == 2 (mod 3), b(p^e) = b(p)*b(p^(e-1)) - b(p^(e-2))*p^3 if p == 1 (mod 3) where b(p) = (x^2 - 3*p)*x, 4*p = x^2 + 3*y^2, |x|<|y| and x == 2 (mod 3). - Michael Somos, Aug 23 2006
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 - u * w * (u + 16 * w). - Michael Somos, Feb 19 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 81 (t/i)^4 f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 29 2011
G.f.: Product_{k>0} (1 - x^k)^8.
a(2*n) = A153728(n). - Michael Somos, Sep 29 2011
a(4*n + 1) = -8 * a(n). - Michael Somos, Dec 06 2004
a(4*n + 3) = a(16*n + 13) = 0. - Michael Somos, Oct 19 2005
A092342(n) = a(n) + 81*A033690(n-1). - Michael Somos, Aug 22 2007
Sum_{n>=0} a(n) * q^(3*n + 1) = (Sum_{i,j,k in Z} (i-j) * (j-k) * (k-i) * q^((i*i + j*j + k*k) / 2)) / 2 where 0 = i+j+k, i == 1 (mod 3), j == 2 (mod 3), and k == 0 (mod 3). - Michael Somos, Sep 22 2014
a(0) = 1, a(n) = -(8/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 27 2017
G.f.: exp(-8*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 2 (mod 3). Then a( M^2*n + (M^2 - 1)/3 ) = (-1)^k*M^3*a(n). See Cooper et al., Theorem 1. - Peter Bala, Dec 01 2020
a(n) = b(3*n + 1) where b(n) is multiplicative and b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * (-p^3)^(e/2) if p == 2 (mod 3), b(p^e) = (((x+sqrt(-3)*y)/2)^(3*e+3) - ((x-sqrt(-3)*y)/2)^(3*e+3))/(((x+sqrt(-3)*y)/2)^3 - ((x-sqrt(-3)*y)/2)^3) if p == 1 (mod 3) where 4*p = x^2 + 3*y^2, |x|<|y| and x == 2 (mod 3). - Jianing Song, Mar 19 2022

Corrected by Charles R Greathouse IV, Sep 02 2009

A286354 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^j)^k.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, -1, 0, 1, -3, -1, 0, 0, 1, -4, 0, 2, 0, 0, 1, -5, 2, 5, 1, 1, 0, 1, -6, 5, 8, 0, 2, 0, 0, 1, -7, 9, 10, -5, 0, -2, 1, 0, 1, -8, 14, 10, -15, -4, -7, 0, 0, 0, 1, -9, 20, 7, -30, -6, -10, 0, -2, 0, 0, 1, -10, 27, 0, -49, 0, -5, 8, 0, -2, 0, 0, 1, -11, 35, -12, -70, 21, 11, 25, 9, 0, 1, 0, 0

Offset: 0

Ilya Gutkovskiy, May 08 2017

A(n,k) number of partitions of n into an even number of distinct parts minus number of partitions of n into an odd number of distinct parts with k types of each part.

			A(3,2) = 2 because we have [2, 1], [2', 1], [2, 1'], [2', 1'] (number of partitions of 3 into an even number of distinct parts with 2 types of each part), [3], [3'] (number of partitions of 3 into an odd number of distinct parts with 2 types of each part) and 4 - 2 = 2.
Square array begins:
1,  1,  1,  1,  1,   1,  ...
0, -1, -2, -3, -4,  -5,  ...
0, -1, -1,  0,  2,   5,  ...
0,  0,  2,  5,  8,  10,  ...
0,  0,  1,  0, -5, -15,  ...
0,  1,  2,  0, -4,  -6,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Columns k=0-20 give: A000007, A010815, A002107, A010816, A000727, A000728, A000729, A000730, A000731, A010817, A010818, A010819, A000735, A010820, A010821, A010822, A000739, A010823, A010824, A010825, A010826.
Main diagonal gives A008705.
Antidiagonal sums give A299105.

A:= proc(n, k) option remember; `if`(n=0, 1, -k*
      add(numtheory[sigma](j)*A(n-j, k), j=1..n)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Jun 21 2018

Table[Function[k, SeriesCoefficient[Product[(1 - x^i)^k , {i, Infinity}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[QPochhammer[x, x, Infinity]^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[(-1)^i*x^(i*(3*i + 1)/2), {i, -Infinity, Infinity}]^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} (1 - x^j)^k.
G.f. of column k: (Sum_{j=-inf..inf} (-1)^j*x^(j*(3*j+1)/2))^k.
Column k is the Euler transform of period 1 sequence [-k, -k, -k, ...].

A000729 Expansion of Product_{k >= 1} (1 - x^k)^6.

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, -294, 0, 0, -510, 378, -540, 138, 0

Offset: 0

N. J. A. Sloane

This is Glaisher's function lambda(m). It appears to be defined only for odd m, and lambda(4t-1) = 0 (t >= 1), lambda(4t+1) = a(t) (t >= 0). - N. J. A. Sloane, Nov 25 2018
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 36 of the 74 eta-quotients listed in Table I of Martin (1996).
Dickson, v.2, p. 295 briefly states a result of Glaisher, 1883, pp 212-215. This result is that a(n) is the sum over all solutions of 16*n + 4 = x^2 + y^2 + z^2 + w^2 in nonnegative odd integers of chi(x) and is also the sum over all solutions of 8*n + 2 = x^2 + y^2 in nonnegative odd integers of chi(x) * chi(y) where chi(x) = x if x == 1 (mod 4) and -x if x == 3 (mod 4). [Michael Somos, Jun 18 2012]
Denoted by g_3(q) in Cynk and Hulek on page 8 as the unique weight 3 Hecke eigenform of level 16 with complex multiplication by i. - Michael Somos, Aug 24 2012
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. - Michael Somos, Aug 24 2012

			G.f. = 1 - 6*x + 9*x^2 + 10*x^3 - 30*x^4 + 11*x^6 + 42*x^7 - 70*x^9 + 18*x^10 + ...
G.f. = q - 6*q^5 + 9*q^9 + 10*q^13 - 30*q^17 + 11*q^25 + 42*q^29 - 70*q^37 + ...

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 295, and vol. 3, p. 134.
J. W. L. Glaisher, On the representations of a number as a sum of four squares, and on some allied arithmetical functions, Quarterly Journal of Pure and Applied Mathematics, 36 (1905), 305-358. See page 340.
J. W. L. Glaisher, The arithmetical functions P(m), Q(m), Omega(m), Quart. J. Math, 37 (1906), 36-48.
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.
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
M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
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. Cynk and K. Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds, arXiv:math/0509424 [math.AG], 2005-2006.
S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
J. W. L. Glaisher, Note on the Compositions of a Number as a Sum of Two and Four Uneven Squares, Quarterly Journal of Pure and Applied Mathematics, 19 (1883), 212-215.
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167.
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167. [Annotated scanned copy]
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 5).
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
S. Milne and V. Leininger, Some new infinite families of eta function identities, Methods and Applications of Analysis 6 (1999), 225--248.
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]
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
Index entries for expansions of Product_{k >= 1} (1-x^k)^m
Index entries for sequences mentioned by Glaisher

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

A := Basis( ModularForms( Gamma1(16), 3), 274); A[2] - 6*A[6] + 9*A[10] + 10*A[14] - 30*A[18]; /* Michael Somos, May 17 2015 */

A := Basis( CuspForms( Gamma1(16), 3), 274); A[1] - 6*A[5]; /* Michael Somos, Jan 09 2017 */

a[ n_] := SeriesCoefficient[ 1/16 EllipticTheta[ 4, 0, q] EllipticTheta[ 2, 0, q]^4 EllipticTheta[ 3, 0, q], {q, 0, 4 n + 1}]; (* Michael Somos, Jun 18 2012 *)
a[ n_] := If[ n < 0, 0, With[ {m = Sqrt[ 16 n + 4]}, SeriesCoefficient[ Sum[ Mod[k, 2] q^k^2, {k, m}]^3 Sum[ KroneckerSymbol[ -4, k] k q^k^2, {k, m}], {q, 0, 16 n + 4}]]]; (* Michael Somos, Jun 12 2012 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ Sqrt[(1 - m) m ] (EllipticK[m] 2/Pi)^3 / (4 q^(1/2)), {q, 0, 2 n}]]; (* Michael Somos, Jun 22 2012 *)
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, n}]^6, {x, 0, n}]; (* Michael Somos, May 17 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^6, {x, 0, n}]; (* Michael Somos, May 17 2015 *)
a[ n_] := SeriesCoefficient[ (-1/4) EllipticThetaPrime[ 1, -Pi/4, q] EllipticTheta[ 1, -Pi/4, q]^3, {q, 0, 4 n + 1}]; (* Michael Somos, May 17 2015 *)
a[ n_] := SeriesCoefficient[ (-1/16) EllipticThetaPrime[ 1, 0, q] EllipticTheta[ 1, -Pi/2, q]^3, {q, 0, 4 n + 1}]; (* Michael Somos, May 17 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( 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^e), forstep( i=1, sqrtint(p), 2, if( issquare( p - i^2, &y), x=i; break)); a0=1; a1 = y = 2*(x^2 - y^2); for( i=2, e, x = y*a1 - p^2*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Aug 21 2006 */

{a(n)=local(tn=(sqrtint(8*n+1)+1)\2);polcoeff(sum(m=0,tn,(1+2*m)^2*x^(m^2+m)+x*O(x^n)) + 2*sum(m=0,tn,sum(k=1,tn,(1+4*(m^2+m-k^2))*x^(m^2+m+k^2)+x*O(x^n))),n)} /* Paul D. Hanna, Mar 15 2010 */

Expansion of q^(-1/4)/16 * theta_2(q)^4 * theta_3(q) * theta_4(q) in powers of q. - [Dickson, v. 3, p. 134] from Stieltjes footnote 160. Michael Somos, Jun 18 2012
Expansion of q^(-1/2) / 4 * k * k' * (K / (pi/2))^3 in powers of q^2 where k, k', K are Jacobi elliptic functions. - Michael Somos, Jun 22 2012
G.f.: Product_{k>0}(1 - x^k)^6.
Given g.f. A(x), then A(q^4) = f(-q^4)^6 = phi(q) * phi(-q) * psi(q^2)^4 where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Aug 23 2006
a(n) = b(4*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = p^e * (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - b(p^(e-2)) * p^2 if p == 1 (mod 4) and b(p) = 2 * (x^2 - y^2) where p = x^2 + y^2 and y is even. - Michael Somos, Aug 23 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 64 (t/i)^3 f(t) where q = exp(2 Pi i t). - Michael Somos, Aug 24 2012
G.f.: Sum_{k>=0} a(k) * x^(4*k + 1) = (1/2) * Sum_{u,v in Z} (u*u - 4*v*v) * x^(u*u + 4*v*v). - Michael Somos, Jun 14 2007
G.f.: eta(x)^6 = Sum_{n>=0} (1+2n)^2*x^(n^2+n) + 2*Sum_{n>=0,k>=1} (1 + 4(n^2+n-k^2))*x^(n^2+n+k^2) - from the Milne and Leininger reference. [Paul D. Hanna, Mar 15 2010]
a(0) = 1, a(n) = -(6/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 26 2017
G.f.: exp(-6*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018
Let M be a positive integer whose prime factors are all congruent to 3 (mod 4) - see A004614. Then a( M^2*n + (M^2 - 1)/4 ) = M^2*a(n). See Cooper et al., equation 5. - Peter Bala, Dec 01 2020
a(n) = b(4*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = p^e * (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = ((x+y*i)^(2*e+2) - (x-y*i)^(2*e+2))/((x+y*i)^2 - (x-y*i)^2) if p == 1 (mod 4) where p = x^2 + y^2 and x is odd. - Jianing Song, Mar 19 2022

A010818 Expansion of Product (1 - x^k)^10 in powers of x.

Original entry on oeis.org

1, -10, 35, -30, -105, 238, 0, -260, -165, 140, 1054, -770, -595, 0, -715, 2162, 455, 0, -2380, -1820, 2401, -680, 1495, 3080, 1615, -6958, -1925, 0, 0, 5100, -1442, 8330, -5355, 1330, 0, -16790, 0, 8190, 8265, 0, 1918, 0, 8415, -10230, -7140, -9362

Offset: 0

N. J. A. Sloane

			G.f. = 1 - 10*x + 35*x^2 - 30*x^3 - 105*x^4 + 238*x^5 - 260*x^7 - 165*x^8 + ...
G.f. = q^5 - 10*q^17 + 35*q^29 - 30*q^41 - 105*q^53 + 238*q^65 - 260*q^89 + ...

Newman, Morris; A table of the coefficients of the powers of eta(tau). Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.

Seiichi Manyama, 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.
S. Cooper, The Quintuple product identity, Int. J. Number Theory 2 (2006), no. 1, 115-161. See after equation (73).
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.

Cf. A122266, A234565.

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

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^10, {x, 0, n}]; (* Michael Somos, Jun 24 2013 *)

{a(n) = if( n<0, 0, polcoeff( eta(x + x * O(x^n))^10, n))}; /* Michael Somos, Jun 09 2011 */

{a(n) = local(m, x, y, z); if( n<0, 0, m = 12*n + 5; z = 0; for( x = -sqrtint(m), sqrtint(m), if( x%6 != 1, next); if( issquare( m - x^2, &y), if( y%6 == 2, y = -y); if( y%6 == 4, z += x*y * (x*x - y*y) ))); z / 6)}; /* Michael Somos, Jun 09 2011 */

{a(n) = local(A, p, e, i, x, y, a0, a1); if( n<0, 0, n = 12*n + 5; A = factor(n); 1 / 48 * prod( k=1, matsize(A)[1], if( p=A[k,1], e=A[k,2]; if( p<5, 0, if( p%12 > 6, if( e%2, 0, p^(2*e)), forstep( i = 1, sqrtint( p), 2, if( issquare( p - i^2, &y), x=i; break)); if( p%12 == 5, a1 = 8 * x*y * (x-y) * (x+y) * (-1)^((x%6==1) + (y%6==4)), a1 = 2 * (x^2-y^2+2*x*y) * (x^2-y^2-2*x*y) * (-1)^(x%6==3) ); a0 = 1; y = a1; for( i=2, e, x = y * a1 - p^4 * a0; a0=a1; a1=x); a1 )))))}; /* Michael Somos, Jun 24 2013 */

Expansion of f(-x)^10 in powers of x where f() is a Ramanujan theta function.
Expansion of q^(-5/12) * eta(q)^10 in powers of q. - Michael Somos, Jun 09 2011
a(n) = b(12*n + 5) / 48 where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * p^(2*e) if p == 7 or 11 (mod 12), b(p^e) = b(p) * b(p^(e-1)) - p^4 * b(p^(e-2)) if p == 1 or 5 (mod 12). - Michael Somos, Jun 24 2013
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 12^5 (t/i)^5 f(t) where q = exp(2 Pi i t). - Michael Somos, Jan 06 2014
G.f.: Product_{k>0} (1 - x^k)^10. a(49*n + 20) = 2401 * a(n).
48 * a(n) = A234565(3*n + 1). a(7*n + 2) = 0 unless n == 2 (mod 7). - Michael Somos, Jul 18 2014
a(0) = 1, a(n) = -(10/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 27 2017
G.f.: exp(-10*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 congruent to 7 (mod 12) or 11 (mod 12). Then a( M^2*n + 10*(M^2 - 1)/24 ) = M^4*a(n). See Cooper et al., Theorem 1. - Peter Bala, Dec 01 2020

A277076 Expansion of f(-x)^8 * Q(x) in powers of x where f() is a Ramanujan theta function and Q() is a Ramanujan Lambert series.

Original entry on oeis.org

1, 232, 260, -5760, 6890, 7744, 33176, -115200, 14035, 60320, 1508, 449280, -380770, -599040, 7640, 599040, -755943, 1598480, 1843200, -2620800, -988858, -2995712, 3857360, -1497600, -2004730, 7696832, 2699684, 1670400, -7188480, -11980800, 1791400, 10736640

Offset: 0

Michael Somos, Sep 27 2016

sign

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

Ramanujan Lambert series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

			G.f. = 1 + 232*x + 260*x^2 - 5760*x^3 + 6890*x^4 + 7744*x^5 + 33176*x^6 - 115200*x^7 + 14035*x^8 + ...
G.f. = q + 232*q^4 + 260*q^7 - 5760*q^10 + 6890*q^13 + 7744*q^16 + 33176*q^19 - 115200*q^22 + ...

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 329, 2nd equation.

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. A000731, A000739, A004009.

A := Basis( CuspForms( Gamma0(9), 8), 95); A[1] + 232*A[4];

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ QPochhammer[ x]^8 (1 + 240 Sum[ DivisorSigma[ 3, k] x^k, {k, n}]), {x, 0, n}]];
a[ n_] := SeriesCoefficient[ With[ {A1 = QPochhammer[ x]^8, A2 = QPochhammer[ x^2]^8}, A1 (A1^3 + 256 x A2^3) / (A1 A2)], {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^8 * sum(k=1, n, 240 * sigma(k, 3) * x^k, 1 + A), n))};

{a(n) = my(A, A1, A2); if( n<0, 0, A = x * O(x^n); A1 = eta(x + A)^8; A2 = eta(x^2 + A)^8; polcoeff( A1 * (A1^3 + 256 * x * A2^3) / (A1 * A2), n))};

Expansion of f(-x)^8 * (f(-x)^24 + 256 * x * f(-x^2)^24) / (f(-x) * f(-x^2))^8 in powers of x.
a(n) = b(3*n+1) where b() is multiplicative with b(p^e) = 0^e if p=3 and b(p^e) = b(p)*b(p^(e-1))  - p^7*b(p^(e-2)) otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 6561 (t/i)^8 f(t) where q = exp(2 Pi i t).
Convolution of A000731 and A004009.

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

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A002107 Expansion of Product_{k>=1} (1 - x^k)^2.

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A000731 Expansion of Product (1 - x^k)^8 in powers of x.

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A286354 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^j)^k.

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A000729 Expansion of Product_{k >= 1} (1 - x^k)^6.

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