A002107 -id:A002107 - cp's OEIS frontend

A258406 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^2 dx.

2, 5, 3, 8, 7, 4, 0, 8, 2, 3, 7, 8, 2, 7, 6, 0, 0, 2, 9, 8, 8, 5, 0, 8, 8, 9, 3, 8, 1, 6, 3, 3, 2, 9, 1, 2, 3, 8, 4, 7, 6, 3, 6, 3, 4, 3, 1, 9, 3, 3, 1, 3, 5, 1, 4, 7, 5, 6, 0, 6, 7, 6, 0, 5, 8, 8, 6, 9, 6, 6, 3, 0, 9, 2, 7, 3, 5, 4, 6, 9, 1, 6, 8, 5, 9, 8, 1, 6, 6, 0, 3, 1, 4, 9, 6, 8, 3, 7, 8, 6, 5, 4, 1, 2, 5, 0

Offset: 0

Vaclav Kotesovec, May 29 2015

			0.2538740823782760029885088938163329123847636343193313514756067...

Vaclav Kotesovec, The integration of q-series

Cf. A002107, A258232, A258407, A258404, A258405.

evalf(Sum(Sum(8*(n+1)*(-1)^n / ((n^2 - 2*k^2 + 2*k*n + n + 2) * (n^2 - 2*k^2 + 2*k*n + 5*n + 6)), k=0..n), n=0..infinity), 120);

RealDigits[NIntegrate[QPochhammer[x]^2, {x, 0, 1}, WorkingPrecision -> 120], 10, 106][[1]] (* Vaclav Kotesovec, Oct 10 2023 *)

Equals Sum_{n>=0} Sum_{k=0..n} 8*(n+1)*(-1)^n / ((n^2 - 2*k^2 + 2*k*n + n + 2) * (n^2 - 2*k^2 + 2*k*n + 5*n + 6)).

Equals Sum_{n>=0} Sum_{j=-floor(n/2)..floor(n/2)} (-1)^(n+j) / (n*(n+1)/2 - j*(3*j-1)/2 + 1).

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

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))

A293388 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} (-1)^jjx^(j*i))^2.

Original entry on oeis.org

1, 1, 0, 1, -2, 0, 1, -2, -1, 0, 1, -2, 3, 2, 0, 1, -2, 3, -2, 1, 0, 1, -2, 3, -8, 1, 2, 0, 1, -2, 3, -8, 7, -6, -2, 0, 1, -2, 3, -8, 15, -6, 14, 0, 0, 1, -2, 3, -8, 15, -14, 17, -20, -2, 0, 1, -2, 3, -8, 15, -24, 17, -14, 22, -2, 0, 1, -2, 3, -8, 15, -24, 27

Offset: 0

Seiichi Manyama, Oct 07 2017

			Square array begins:
   1,  1,  1,  1,   1, ...
   0, -2, -2, -2,  -2, ...
   0, -1,  3,  3,   3, ...
   0,  2, -2, -8,  -8, ...
   0,  1,  1,  7,  15, ...
   0,  2, -6, -6, -14, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..1 give A000007, A002107.
Rows n=0 gives A000012.
Main diagonal gives A293389.
Product_{i>0} 1/(1 + Sum_{j=1..k} (-1)^j*j*x^(j*i))^m: A292577 (m=-2), A293307 (m=-1), A293305 (m=1), this sequence (m=2).

A369710 Maximal coefficient of (1 - x)^2 * (1 - x^2)^2 * (1 - x^3)^2 * ... * (1 - x^n)^2.

Original entry on oeis.org

1, 1, 4, 3, 10, 6, 20, 12, 34, 24, 64, 52, 116, 103, 208, 223, 410, 470, 808, 992, 1620, 2120, 3352, 4494, 6980, 9584, 14680, 20400, 31128, 43774, 66288, 93968, 141654, 201766, 303716, 433746, 652612, 936334, 1404920, 2021344, 3029564, 4364300, 6541872, 9437054

Offset: 0

Ilya Gutkovskiy, Jan 29 2024

nonn

Cf. A002107, A047653, A086376.

Table[Max[CoefficientList[Product[(1 - x^k)^2, {k, 1, n}], x]], {n, 0, 43}]

a(n) = vecmax(Vec(prod(k=1, n, (1-x^k)^2))); \\ Michel Marcus, Jan 30 2024

A122266 Expansion of f(-q)^2 * Q(q) in powers of q.

Original entry on oeis.org

1, 238, 1679, 2162, 2401, -6958, -1442, -23040, 1918, -9362, 14641, 0, 80640, -20398, 28083, 64078, -38398, -69120, 0, -90482, -58562, 0, -241920, 100558, 146879, 0, -193438, 399602, 104638, 114002, 130321, 24242, 0, 107282, -276962, 351118

Offset: 0

Michael Somos, Aug 28 2006

sign

f(-q) (g.f. A010815) and Q(q) (g.f. A004009) are Ramanujan q-series.

			G.f. = 1 + 238*x + 1679*x^2 + 2162*x^3 + 2401*x^4 - 6958*x^5 - 1442*x^6 + ...
G.f. = q + 238*q^13 + 1679*q^25 + 2162*q^37 + 2401*q^49 - 6958*q^61 - 1442*q^73 + ...

Steven R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.

Cf. A002107, A004009, A010818.

a[ n_] := SeriesCoefficient[ (1 + 240 Sum[ DivisorSigma[ 3, k] q^k, {k, n}]) QPochhammer[ q]^2, {q, 0, n}]; (* Michael Somos, Jun 24 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 (EllipticTheta[ 4, 0, q]^8 + EllipticTheta[ 2, 0, q^(1/2)]^8), {q, 0, n}]; (* Michael Somos, Jun 25 2013 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * sum( k=1, n, 240 * sigma( k, 3) * x^k, 1 + A), n))};

{a(n) = my(A, B); if( n<0, 0, A = x * O(x^n); B = 64 * x * (eta(x^4 + A) / eta(x + A))^8; polcoeff( (1 + 4*B + B^2) * eta(x + A)^18 / eta(x^2 + A)^8, n))}; /* Michael Somos, Jun 25 2013 */

{a(n) = my(A, p, e, i, x, y, a0, a1); if( n<0, 0, n = 12*n + 1; A = factor(n); 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 q^(-1/12) * (eta(q)^16 + 256 * eta(q)^8 * eta(q^4)^8 + 4096 * eta(q^4)^16) * eta(q)^2 / eta(q^2)^8 in powers of q. - Michael Somos, Jun 25 2013
a(n) = b(12*n + 1) 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
Convolution of A002107 and A004009. - Michael Somos, Jun 25 2013
Expansion of q^(-1/12) * (eta(q)^24 + 256*eta(q^2)^24) / (eta(q)^6*eta(q^2)^8) = q^(-1/12) * (eta(q)^12 + 250*eta(q)^6*eta(q^5)^6 + 3125*eta(q^5)^12) / eta(q^5)^2 in powers of q. - Michael Somos, Feb 03 2023

A208845 Expansion of f(x)^2 in powers of x where f() is a Ramanujan theta function.

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, 2

Offset: 0

Michael Somos, Mar 03 2012

sign

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

Number 73 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 + ...

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. A002107, A209941.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^2 , {x, 0, n}]; (* Michael Somos, Jun 09 2015 *)

{a(n) = my(A, p, e, m); 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), m=i; break)); (e+1) * (-1)^(e * ( (p%24>1) + m )))))};

{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)))^2, n))};

Expansion of q^(-1/12) * (eta(q^2)^3 / (eta(q) * eta(q^4)))^2 in powers of q.
Euler transform of period 4 sequence [ 2, -4, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = (24)^(1/2) (t/i)^(1/2) f(t) where q = exp(2 Pi i t).
a(n) = b(12*n + 1) where b(n) is multiplicative with  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 * ((p%24>1) + x)) if p == 1 (mod 12) and p = x^2 + 9 * y^2.
a(n) = (-1)^n * A002107(n). a(25*n + 2) = -a(n).
Convolution cube is A209941. - Michael Somos, Jun 09 2015

A304080 Triangular array T(n,k) giving coefficients in expansion of Product_{j=1..n} (1-x^j)^2.

Original entry on oeis.org

1, 1, -2, 1, 1, -2, -1, 4, -1, -2, 1, 1, -2, -1, 2, 3, 0, -6, 0, 3, 2, -1, -2, 1, 1, -2, -1, 2, 1, 4, -4, -4, -2, 0, 10, 0, -2, -4, -4, 4, 1, 2, -1, -2, 1, 1, -2, -1, 2, 1, 2, 0, -2, -6, -2, 3, 6, 5, 2, -3, -12, -3, 2, 5, 6, 3, -2, -6, -2, 0, 2, 1, 2, -1, -2, 1

Offset: 0

Seiichi Manyama, May 06 2018

			Irregular triangle starts:
n\k| 0   1   2  3   4   5   6   7   8  9  10  11  12  13  14 15 16 17  18  19 20
---+-----------------------------------------------------------------------------
0  | 1;
1  | 1, -2,  1;
2  | 1, -2, -1, 4, -1, -2,  1;
3  | 1, -2, -1, 2,  3,  0, -6,  0,  3, 2, -1, -2,  1;
4  | 1, -2, -1, 2,  1,  4, -4, -4, -2, 0, 10,  0, -2, -4, -4, 4, 1, 2, -1, -2, 1;

Seiichi Manyama, Rows n = 0..30, flattened

Cf. A002107, A231599, A303992.

T(n, k) = polcoef(prod(j=1, n, (1-x^j)^2), k);
tabf(nn) = for(n=0, nn, for(k=0, n*(n+1), print1(T(n, k), ", ")); print)

A371551 Expansion of e.g.f. Product_{k>=1} (1 - x^k/k!)^2.

Original entry on oeis.org

1, -2, 0, 10, -4, -42, -258, 306, 5980, 3142, 61730, -794334, -3299074, -8459830, 40220390, 1550926110, 1631691740, 43693916390, -125593997262, -4079362135854, -32054212967294, -33715330874838, -600410923342450, 9383532800084966, 329821022627776798

Offset: 0

Ilya Gutkovskiy, Mar 27 2024

sign

Cf. A002107, A032312, A185895, A371552.

nmax = 24; CoefficientList[Series[Product[(1 - x^k/k!)^2, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

A380499 Absolute value of the minimum coefficient of (1 - x)^2 * (1 - x^2)^2 * (1 - x^3)^2 * ... * (1 - x^n)^2.

Original entry on oeis.org

1, 2, 2, 6, 4, 12, 8, 24, 19, 44, 36, 78, 74, 148, 156, 286, 322, 556, 682, 1120, 1448, 2308, 3072, 4784, 6538, 10064, 14001, 21296, 29928, 45276, 64032, 96712, 137520, 207156, 296236, 444748, 637812, 956884, 1373622, 2062080, 2968872, 4450120, 6422472, 9616202, 13894990, 20802836

Offset: 0

Ilya Gutkovskiy, Jan 25 2025

nonn

Cf. A002107, A047653, A086394, A133871, A369710.

p:= proc(n) option remember;
     `if`(n=0, 1, expand(p(n-1)*(1-x^n)^2))
    end:
a:= n-> abs(min(coeffs(p(n)))):
seq(a(n), n=0..45);  # Alois P. Heinz, Jan 25 2025

Table[Min[CoefficientList[Product[(1 - x^k)^2, {k, 1, n}], x]], {n, 0, 45}] // Abs

a(n) = abs(vecmin(Vec(prod(k=1, n, (1-x^k)^2)))); \\ Michel Marcus, Jan 25 2025

cp's OEIS Frontend

A258406 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^2 dx.

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

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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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Julia

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A293388 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} (-1)^j*j*x^(j*i))^2.

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A369710 Maximal coefficient of (1 - x)^2 * (1 - x^2)^2 * (1 - x^3)^2 * ... * (1 - x^n)^2.

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A122266 Expansion of f(-q)^2 * Q(q) in powers of q.

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A208845 Expansion of f(x)^2 in powers of x where f() is a Ramanujan theta function.

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A293388 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} (-1)^jjx^(j*i))^2.