A047638 -id:A047638 - cp's OEIS frontend

A001482 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^4 in powers of x.

1, -4, 6, -4, -3, 12, -16, 16, -6, -8, 18, -28, 26, -20, 2, 12, -23, 32, -36, 28, -6, 4, 22, -20, 39, -32, 32, -12, 2, 16, -12, 24, -40, 28, -34, 0, -6, -16, 0, -40, 6, -36, 26, -32, -5, 0, -20, 8, -16, 12, -10, 40, -22, 12, 14, 12, 45, 16, 38, 4, 12, 0, 34, 8, 38, 12, -24, 44, 2, 16

Offset: 4

N. J. A. Sloane

sign

H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
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 = 4..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)

Cf. A001483 - A001488, A047638 - A047649, A047654, A047655, A341243.

m:=102;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( (1 - (&*[1-(-x)^j: j in [1..m+2]]))^4 )); // G. C. Greubel, Sep 04 2023

g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 4):
seq(a(n), n=4..73);  # Alois P. Heinz, Feb 07 2021

nmax = 73; CoefficientList[Series[(Product[(1 - (-x)^j), {j, 1, nmax}] - 1)^4, {x, 0, nmax}], x] // Drop[#, 4] & (* Ilya Gutkovskiy, Feb 07 2021 *)
Drop[CoefficientList[Series[(1 -QPochhammer[-x])^4, {x,0,100}], x], 4] (* G. C. Greubel, Sep 04 2023 *)

my(N=70,x='x+O('x^N)); Vec((eta(-x)-1)^4) \\ Joerg Arndt, Sep 04 2023

m=100
def f4(x): return (1 - product( (1+x^j)*(1-x^(2*j))/(1+x^(2*j)) for j in range(1,m+2) ) )^4
def A001482_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f4(x) ).list()
a=A001482_list(m); a[4:] # G. C. Greubel, Sep 04 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A001488 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^10 in powers of x.

Original entry on oeis.org

1, -10, 45, -120, 200, -162, -160, 810, -1530, 1730, -749, -1630, 4755, -7070, 6700, -2450, -5295, 14070, -20010, 19350, -10157, -6290, 25515, -40660, 44940, -34268, 9180, 24510, -57195, 78060, -79087, 56610, -13935, -39600, 89805, -121638, 125405

Offset: 10

N. J. A. Sloane

sign

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 = 10..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)

Cf. A001482 - A001487, A001490, A047638 - A047649, A047654, A047655, A341243.

m:=102;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^10 )); // G. C. Greubel, Sep 04 2023

g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 10):
seq(a(n), n=10..46);  # Alois P. Heinz, Feb 07 2021

nmax=46; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] -1)^10, {x,0,nmax}], x]//Drop[#,10] & (* Ilya Gutkovskiy, Feb 07 2021 *)
Drop[CoefficientList[Series[(QPochhammer[-x] -1)^10, {x,0,102}], x],10] (* G. C. Greubel, Sep 04 2023 *)

my(N=55,x='x+O('x^N)); Vec((eta(-x)-1)^10) \\ Joerg Arndt, Sep 05 2023

from sage.modular.etaproducts import qexp_eta
m=100; k=10;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A001488_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A001488_list(m); a[k:] # G. C. Greubel, Sep 04 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^10. - G. C. Greubel, Sep 04 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A047654 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^2 in powers of x.

Original entry on oeis.org

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

Offset: 2

N. J. A. Sloane

sign

Alois P. Heinz, Table of n, a(n) for n = 2..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.

Cf. A001482 - A001488, A001490, A047265, A047638 - A047649, A047655, A341243.

m:=120;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^2 )); // G. C. Greubel, Sep 07 2023

g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 2):
seq(a(n), n=2..94);  # Alois P. Heinz, Feb 07 2021

nmax=94; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] - 1)^2, {x, 0, nmax}], x]//Drop[#, 2] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=2}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 125}], x], k]] (* G. C. Greubel, Sep 07 2023 *)

seq(n)={Vec((prod(j=1, n, 1-(-x)^j + O(x^n)) - 1)^2)} \\ Andrew Howroyd, Feb 07 2021

from sage.modular.etaproducts import qexp_eta
m=125; k=2;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047654_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047654_list(m); a[k:] # G. C. Greubel, Sep 07 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^2. - G. C. Greubel, Sep 07 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A047655 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^3 in powers of x.

Original entry on oeis.org

1, -3, 3, -1, -3, 6, -6, 6, 0, -3, 6, -9, 8, -6, 0, 0, -6, 6, -13, 3, -6, 3, 0, -3, 6, -9, 6, -3, 6, 0, 6, 6, -3, 11, 0, 6, 0, 9, 0, 0, 0, -3, 13, 0, 0, -6, 0, -6, 3, -3, -6, 0, -15, -6, -3, 0, -6, 0, -6, 0, -6, -6, 0, -11, 0, 0, -6, 0, 6, 0, 6, 0, 0, 0, -3, 19, 12, -3, 0, 0, 6, 6, 6, 6, 0, 0, 6, 0, 21, 3

Offset: 3

N. J. A. Sloane

sign

Alois P. Heinz, Table of n, a(n) for n = 3..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.

Cf. A001482 - A001488, A001490, A047265, A047638 - A047649, A047654, A341243.

m:=120;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^3 )); // G. C. Greubel, Sep 07 2023

g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 3):
seq(a(n), n=3..92);  # Alois P. Heinz, Feb 07 2021

nmax=92; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] - 1)^3, {x,0,nmax}], x]//Drop[#, 3] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=3}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 125}], x], k]] (* G. C. Greubel, Sep 07 2023 *)

my(x='x+O('x^99)); Vec((eta(-x)-1)^3) \\ Joerg Arndt, Sep 07 2023

from sage.modular.etaproducts import qexp_eta
m=125; k=3;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047655_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047655_list(m); a[k:] # G. C. Greubel, Sep 07 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^3. - G. C. Greubel, Sep 07 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A047649 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^11 in powers of x.

Original entry on oeis.org

1, -11, 55, -165, 319, -352, -44, 1100, -2585, 3542, -2519, -1530, 8085, -14410, 16170, -9460, -6644, 28105, -46145, 50248, -32802, -6193, 57200, -102575, 121968, -100397, 35123, 60390, -158840, 226413, -234344, 168773, -37070, -131175, 290851, -391402

Offset: 11

N. J. A. Sloane

sign

Alois P. Heinz, Table of n, a(n) for n = 11..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.

Cf. A001482 - A001488, A001490, A047638 - A047648, A047654, A047655, A341243.

m:=75;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(11) )); // G. C. Greubel, Sep 05 2023

g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 11):
seq(a(n), n=11..46);  # Alois P. Heinz, Feb 07 2021

nmax=46; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] - 1)^11, {x,0,nmax}], x]//Drop[#, 11] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=11}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 75}], x], k]] (* G. C. Greubel, Sep 05 2023 *)

my(N=55,x='x+O('x^N)); Vec((eta(-x)-1)^11) \\ Joerg Arndt, Sep 05 2023

from sage.modular.etaproducts import qexp_eta
m=75; k=11;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047649_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047649_list(m); a[k:] # G. C. Greubel, Sep 05 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^11. - G. C. Greubel, Sep 05 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A001490 Expansion of {Product_{j>=1} (1 - (-x)^j) - 1}^12 in powers of x.

Original entry on oeis.org

1, -12, 66, -220, 483, -660, 252, 1320, -4059, 6644, -6336, 240, 12255, -27192, 35850, -27972, -2343, 50568, -99286, 122496, -96162, 11584, 115116, -242616, 315216, -283800, 128304, 126280, -409398, 622644, -671550, 501468, -122508, -382360

Offset: 1

N. J. A. Sloane

sign

H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
M. Kontsevich and D. Zagier, Periods, pp. 771-808 of B. Engquist and W. Schmid, editors, Mathematics Unlimited - 2001 and Beyond, 2 vols., Springer-Verlag, 2001.
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).

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001482 - A001488, A047638 - A047649, A047654, A047655, A341243.

m:=102;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(12) )); // G. C. Greubel, Sep 05 2023

With[{k=12}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x, 0, 102}], x], k]] (* G. C. Greubel, Sep 04 2023 *)

my(N=55,x='x+O('x^N)); Vec((eta(-x)-1)^12) \\ Joerg Arndt, Sep 05 2023

from sage.modular.etaproducts import qexp_eta
m=100; k=12;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A001490_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A001490_list(m); a[k:] # G. C. Greubel, Sep 05 2023

G.f.: (eta(z)*eta(6*z)/(eta(2*z)*eta(3*z)))^12.

a(n) = [x^n]( QPochhammer(-x) - 1 )^12. - G. C. Greubel, Sep 05 2023

A001483 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^5 in powers of x.

Original entry on oeis.org

1, -5, 10, -10, 0, 19, -35, 40, -25, -10, 45, -75, 80, -60, 15, 45, -85, 115, -115, 90, -21, -35, 95, -130, 135, -135, 70, -35, -65, 105, -146, 120, -150, 90, -65, -25, 90, -115, 150, -125, 130, -45, 80, 35, -5, 160, -110, 170, -85, 95, 25, 50, 0, -60, 95, -116, 120, -135

Offset: 5

N. J. A. Sloane

sign

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 = 5..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)

Cf. A001482, A001484 - A001488, A047638 - A047649, A047654, A047655, A341243.

m:=102;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^5 )); // G. C. Greubel, Sep 04 2023

g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 5):
seq(a(n), n=5..62);  # Alois P. Heinz, Feb 07 2021

nmax = 62; CoefficientList[Series[(Product[(1 - (-x)^j), {j, 1, nmax}] - 1)^5, {x, 0, nmax}], x] // Drop[#, 5] & (* Ilya Gutkovskiy, Feb 07 2021 *)
Drop[CoefficientList[Series[(QPochhammer[-x] -1)^5, {x,0,102}], x], 5] (* G. C. Greubel, Sep 04 2023 *)

my(N=70,x='x+O('x^N)); Vec((eta(-x)-1)^5) \\ Joerg Arndt, Sep 04 2023

m=100; k=5;
def f(k,x): return (-1 + product( (1+x^j)*(1-x^(2*j))/(1+x^(2*j)) for j in range(1,m+2) ) )^k
def A001483_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A001483_list(m); a[k:] # G. C. Greubel, Sep 04 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^5. - G. C. Greubel, Sep 04 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A001487 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^9 in powers of x.

Original entry on oeis.org

1, -9, 36, -84, 117, -54, -177, 540, -837, 755, -54, -1197, 2535, -3204, 2520, -246, -3150, 6426, -8106, 7011, -2844, -3549, 10359, -15120, 15804, -11403, 2574, 8610, -18972, 25425, -25824, 18954, -6165, -10080, 25101, -35262, 37799, -31374, 17379, 1929

Offset: 9

N. J. A. Sloane

sign

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 = 9..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)

Cf. A001482 - A001486, A001488, A047638 - A047649, A047654, A047655, A341243.

m:=102;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^9 )); // G. C. Greubel, Sep 04 2023

g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 9):
seq(a(n), n=9..48);  # Alois P. Heinz, Feb 07 2021

nmax=48; CoefficientList[Series[(Product[(1 - (-x)^j), {j,nmax}] -1)^9, {x,0,nmax}], x]//Drop[#,9] & (* Ilya Gutkovskiy, Feb 07 2021 *)
Drop[CoefficientList[Series[(QPochhammer[-x] -1)^9, {x,0,102}], x], 9] (* G. C. Greubel, Sep 04 2023 *)

my(N=55,x='x+O('x^N)); Vec((eta(-x)-1)^9) \\ Joerg Arndt, Sep 05 2023

from sage.modular.etaproducts import qexp_eta
m=100; k=9;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A001487_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A001487_list(m); a[k:] # G. C. Greubel, Sep 04 2023

a(n) = [x^n]( QPochhammer(-x) - 1 )^9. - G. C. Greubel, Sep 04 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A001484 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^6 in powers of x.

Original entry on oeis.org

1, -6, 15, -20, 9, 24, -65, 90, -75, 6, 90, -180, 220, -180, 66, 110, -264, 360, -365, 264, -66, -178, 375, -510, 496, -414, 180, 60, -330, 570, -622, 582, -390, 220, 96, -300, 621, -630, 705, -492, 300, 0, -235, 420, -570, 594, -735, 420, -420, -120, 219, -586, 360

Offset: 6

N. J. A. Sloane

sign

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).

Robert Israel, Table of n, a(n) for n = 6..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)

Cf. A001482, A001483, A001485 - A001488, A047638 - A047649, A047654, A047655, A341243.

m:=102;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^6 )); // G. C. Greubel, Sep 04 2023

N:= 100:
S:= series((mul(1-(-x)^j,j=1..N)-1)^6,x,N+1):
seq(coeff(S,x,j),j=6..N); # Robert Israel, Feb 05 2019

Drop[CoefficientList[Series[(QPochhammer[-x] -1)^6, {x,0,102}], x], 6] (* G. C. Greubel, Sep 04 2023 *)

my(N=70,x='x+O('x^N)); Vec((eta(-x)-1)^6) \\ Joerg Arndt, Sep 04 2023

m=100; k=6;
def f(k,x): return (-1 + product( (1+x^j)*(1-x^(2*j))/(1+x^(2*j)) for j in range(1,m+2) ) )^k
def A001484_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A001484_list(m); a[k:] # G. C. Greubel, Sep 04 2023

a(n) = [x^n] ( QPochhammer(-x) - 1 )^6. - G. C. Greubel, Sep 04 2023

Edited by Robert Israel, Feb 05 2019

A001485 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^7 in powers of x.

Original entry on oeis.org

1, -7, 21, -35, 28, 21, -105, 181, -189, 77, 140, -385, 546, -511, 252, 203, -693, 1029, -1092, 798, -203, -581, 1281, -1708, 1687, -1232, 413, 602, -1485, 2233, -2366, 2009, -1099, 14, 1099, -2072, 2667, -2807, 2254, -1477, 0, 1057, -2346, 2744, -3017, 2457

Offset: 7

N. J. A. Sloane

sign

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 = 7..10000
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)

Cf. A001482, A001483, A001484, A001486, A001487, A001488, A047638 - A047649, A047654, A047655, A341243.

m:=102;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^7 )); // G. C. Greubel, Sep 04 2023

g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 7):
seq(a(n), n=7..52);  # Alois P. Heinz, Feb 07 2021

nmax = 52; CoefficientList[Series[(Product[(1 - (-x)^j), {j, 1, nmax}] - 1)^7, {x, 0, nmax}], x] // Drop[#, 7] & (* Ilya Gutkovskiy, Feb 07 2021 *)
Drop[CoefficientList[Series[(QPochhammer[-x] -1)^7, {x,0,102}], x], 7] (* G. C. Greubel, Sep 04 2023 *)

my(N=70,x='x+O('x^N)); Vec((eta(-x)-1)^7) \\ Joerg Arndt, Sep 04 2023

m=100; k=7;
def f(k,x): return (-1 + product( (1+x^j)*(1-x^(2*j))/(1+x^(2*j)) for j in range(1,m+2) ) )^k
def A001485_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A001485_list(m); a[k:] # G. C. Greubel, Sep 04 2023

a(n) = [x^n] ( QPochhammer(-x) - 1 )^7. - G. C. Greubel, Sep 04 2023

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

cp's OEIS Frontend

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