cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 21-24 of 24 results.

A047646 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^21 in powers of x.

Original entry on oeis.org

1, -21, 210, -1330, 5964, -19929, 50253, -91920, 97965, 51604, -526659, 1389297, -2280320, 2118690, 769065, -7613319, 17220042, -23999430, 18024405, 10748850, -63778953, 124134772, -152793270, 99072120, 71722224, -341062407, 610085721
Offset: 21

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A001482 - A001488, A001490, A047638 - A047645, A047647 - A047649, A047654, A047655, A341243.

Programs

  • Magma
    m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(21) )); // G. C. Greubel, Sep 06 2023
  • Maple
    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, 21):
seq(a(n), n=21..47);  # Alois P. Heinz, Feb 07 2021
  • Mathematica
    nmax=47; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] -1)^21, {x, 0, nmax}], x]//Drop[#, 21] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=21}, Drop[CoefficientList[Series[(QPochhammer[-x] - 1)^k, {x,0, 75}], x], k]] (* G. C. Greubel, Sep 06 2023 *)
  • PARI
    my(N=44, x='x+O('x^N)); Vec((eta(-x)-1)^21) \\ Joerg Arndt, Sep 06 2023
  • SageMath
    from sage.modular.etaproducts import qexp_eta
m=75; k=21;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047646_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047646_list(m); a[k:] # G. C. Greubel, Sep 06 2023

Formula

a(n) = [x^n]( QPochhammer(-x) - 1 )^21. - G. C. Greubel, Sep 06 2023

Extensions

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A047647 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^22 in powers of x.

Original entry on oeis.org

1, -22, 231, -1540, 7293, -25872, 69971, -140822, 183711, -25102, -634480, 2027804, -3817814, 4439116, -919600, -9829270, 27660479, -44779042, 43632974, -1898820, -92518261, 219961214, -313463842, 267448104, 15757973, -547042056, 1173033400
Offset: 22

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A001482 - A001488, A001490, A047638 - A047646, A047648, A047649, A047654, A047655, A341243.

Programs

  • Magma
    m:=75;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(22) )); // G. C. Greubel, Sep 05 2023
  • Maple
    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, 22):
seq(a(n), n=22..48);  # Alois P. Heinz, Feb 07 2021
  • Mathematica
    nmax=48; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] - 1)^22, {x, 0, nmax}], x]//Drop[#, 22] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=22}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 75}], x], k]] (* G. C. Greubel, Sep 05 2023 *)
  • PARI
    my(N=55, x='x+O('x^N)); Vec((eta(-x)-1)^22) \\ Michel Marcus, Sep 05 2023
  • SageMath
    from sage.modular.etaproducts import qexp_eta
m=75; k=22;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047647_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047647_list(m); a[k:] # G. C. Greubel, Sep 05 2023

Formula

a(n) = [x^n]( QPochhammer(-x) - 1 )^22. - G. C. Greubel, Sep 05 2023

Extensions

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

A341253 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^10.

Original entry on oeis.org

1, 0, 10, 10, 55, 100, 265, 560, 1175, 2420, 4667, 9000, 16575, 30180, 53470, 93152, 159395, 268190, 444910, 727360, 1174563, 1873320, 2955010, 4611960, 7127305, 10912244, 16560430, 24924550, 37217620, 55160650, 81174270, 118651560, 172316445, 248718830, 356892660
Offset: 10

Views

Author

Ilya Gutkovskiy, Feb 07 2021

Keywords

Crossrefs

Cf. A000700, A001488, A022605, A327388, A338463, A341236, A341241, A341243, A341244, A341245, A341246, A341247, A341251.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -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<2, `if`(n=0, 1-k, 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..44);  # Alois P. Heinz, Feb 07 2021
  • Mathematica
    nmax = 44; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^10, {x, 0, nmax}], x] // Drop[#, 10] &

Formula

G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^10.

A341263 Coefficient of x^(2*n) in (-1 + Product_{k>=1} (1 - x^k))^n.

Original entry on oeis.org

1, -1, 1, -1, -3, 19, -65, 181, -419, 755, -749, -1530, 12255, -47477, 141065, -343526, 660941, -770917, -911369, 9721976, -40135713, 124134772, -313463842, 631382751, -824406065, -492101356, 8192253811, -35948431288, 115087580857, -299576625051, 627027769120, -894734468883
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 07 2021

Keywords

Crossrefs

Cf. A001482, A001483, A001484, A001485, A001486, A001487, A001488, A008705, A010815, A047654, A047655, A324595, A340987, A341265.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n=0, 1, add(add(
     -d, 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, g(n+1),
      (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=0..31);  # Alois P. Heinz, Feb 07 2021
  • Mathematica
    Table[SeriesCoefficient[(-1 + QPochhammer[x, x])^n, {x, 0, 2 n}], {n, 0, 31}]
A[n_, k_] := A[n, k] = If[n == 0, 1, -k Sum[A[n - j, k] DivisorSigma[1, j], {j, 1, n}]/n]; T[n_, k_] := Sum[(-1)^i Binomial[k, i] A[n, k - i], {i, 0, k}];
Table[T[2 n, n], {n, 0, 31}]
Previous Showing 21-24 of 24 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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