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.

Showing 1-3 of 3 results.

A047644 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^19 in powers of x.

Original entry on oeis.org

1, -19, 171, -969, 3857, -11286, 24206, -34542, 14706, 83011, -294880, 569753, -680694, 220286, 1198672, -3502612, 5661867, -5571579, 791350, 9721976, -23494393, 33415357, -29225230, 2352751, 47086598, -104517176, 140834118, -121255530
Offset: 19

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A001482 - A001488, A001490, A047638 - A047643, A047645 - A047649, A047654, A047655, A341243.

Programs

  • Magma
    m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(19) )); // G. C. Greubel, Sep 07 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, 19):
seq(a(n), n=19..46);  # Alois P. Heinz, Feb 07 2021
  • Mathematica
    nmax=46; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] - 1)^19, {x,0,nmax}], x]//Drop[#, 19] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=19}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 75}], x], k]] (* G. C. Greubel, Sep 07 2023 *)
  • PARI
    my(x='x+O('x^35)); Vec((eta(-x)-1)^19) \\ Joerg Arndt, Sep 07 2023
  • SageMath
    from sage.modular.etaproducts import qexp_eta
m=75; k=19;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047644_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047644_list(m); a[k:] # G. C. Greubel, Sep 07 2023

Formula

a(n) = [x^n]( QPochhammer(-x) - 1 )^19. - G. C. Greubel, Sep 07 2023

Extensions

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

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

A047444 Numbers that are congruent to {0, 3, 5, 6} mod 8.

Original entry on oeis.org

0, 3, 5, 6, 8, 11, 13, 14, 16, 19, 21, 22, 24, 27, 29, 30, 32, 35, 37, 38, 40, 43, 45, 46, 48, 51, 53, 54, 56, 59, 61, 62, 64, 67, 69, 70, 72, 75, 77, 78, 80, 83, 85, 86, 88, 91, 93, 94, 96, 99, 101, 102, 104, 107, 109, 110, 112, 115, 117, 118, 120, 123, 125
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047398, A047645.

Programs

  • Magma
    [n : n in [0..150] | n mod 8 in [0, 3, 5, 6]]; // Wesley Ivan Hurt, May 26 2016
  • Maple
    A047444:=n->(1+I)*(4*n-4*n*I+3*I-3-I^(-n)+I^(1+n))/4: seq(A047444(n), n=1..100); # Wesley Ivan Hurt, May 26 2016
  • Mathematica
    Table[(1+I)*(4n-4*n*I+3*I-3-I^(-n)+I^(1+n))/4, {n, 80}] (* Wesley Ivan Hurt, May 26 2016 *)
LinearRecurrence[{2,-2,2,-1},{0,3,5,6},70] (* Harvey P. Dale, Aug 26 2019 *)

Formula

G.f.: x^2*(3-x+2*x^2) / ( (x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 07 2011
From Wesley Ivan Hurt, May 26 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.
a(n) = (1+i)*(4*n-4*n*i+3*i-3-i^(-n)+i^(1+n))/4 where i=sqrt(-1).
a(2k) = A047398(k), a(2k-1) = A047645(k). (End)
E.g.f.: (4 - sin(x) - cos(x) + (4*x - 3)*exp(x))/2. - Ilya Gutkovskiy, May 27 2016
Sum_{n>=2} (-1)^n/a(n) = 3*log(2)/8 - (3-2*sqrt(2))*Pi/16. - Amiram Eldar, Dec 21 2021
Showing 1-3 of 3 results.

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

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