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 1-4 of 4 results.

A341245 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^6.

Original entry on oeis.org

1, 0, 6, 6, 21, 36, 71, 132, 222, 392, 633, 1038, 1629, 2544, 3885, 5842, 8691, 12738, 18494, 26520, 37722, 53132, 74235, 102882, 141579, 193506, 262713, 354552, 475749, 634932, 842922, 1113630, 1464450, 1917254, 2499330, 3244998, 4196966, 5408004, 6943632, 8884996
Offset: 6

Views

Author

Ilya Gutkovskiy, Feb 07 2021

Keywords

Crossrefs

Cf. A000700, A001484, A022601, A112150, A327384, A338463, A341225, A341241, A341243, A341244, 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, 6):
seq(a(n), n=6..45);  # Alois P. Heinz, Feb 07 2021
  • Mathematica
    nmax = 45; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^6, {x, 0, nmax}], x] // Drop[#, 6] &

Formula

G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^6.

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

Views

Author

N. J. A. Sloane

Keywords

References

  • 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).

Links

Crossrefs

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

Programs

  • Magma
    m:=102;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^5 )); // G. C. Greubel, Sep 04 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, 5):
seq(a(n), n=5..62);  # Alois P. Heinz, Feb 07 2021
  • Mathematica
    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 *)
  • PARI
    my(N=70,x='x+O('x^N)); Vec((eta(-x)-1)^5) \\ Joerg Arndt, Sep 04 2023
  • SageMath
    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

Formula

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

Extensions

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

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

Views

Author

N. J. A. Sloane

Keywords

References

  • 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).

Links

Crossrefs

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

Programs

  • Magma
    m:=102;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^7 )); // G. C. Greubel, Sep 04 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, 7):
seq(a(n), n=7..52);  # Alois P. Heinz, Feb 07 2021
  • Mathematica
    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 *)
  • PARI
    my(N=70,x='x+O('x^N)); Vec((eta(-x)-1)^7) \\ Joerg Arndt, Sep 04 2023
  • SageMath
    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

Formula

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

Extensions

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

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}]
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.