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.

A341246 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^7.

Original entry on oeis.org

1, 0, 7, 7, 28, 49, 105, 203, 364, 672, 1141, 1960, 3220, 5250, 8359, 13104, 20272, 30877, 46522, 69195, 101941, 148604, 214697, 307475, 436849, 615965, 862246, 1199009, 1656642, 2275231, 3106824, 4219502, 5701066, 7664923, 10256771, 13663574, 18123924, 23941190
Offset: 7

Views

Author

Ilya Gutkovskiy, Feb 07 2021

Keywords

Crossrefs

Cf. A000700, A001485, A022602, A327385, A338463, A341226, A341241, A341243, A341244, A341245, 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, 7):
seq(a(n), n=7..44);  # Alois P. Heinz, Feb 07 2021
  • Mathematica
    nmax = 44; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^7, {x, 0, nmax}], x] // Drop[#, 7] &

Formula

G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^7.

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

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, A001485 - 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)^6 )); // G. C. Greubel, Sep 04 2023
  • Maple
    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
  • Mathematica
    Drop[CoefficientList[Series[(QPochhammer[-x] -1)^6, {x,0,102}], x], 6] (* G. C. Greubel, Sep 04 2023 *)
  • PARI
    my(N=70,x='x+O('x^N)); Vec((eta(-x)-1)^6) \\ Joerg Arndt, Sep 04 2023
  • SageMath
    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

Formula

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

Extensions

Edited by Robert Israel, Feb 05 2019

A001486 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^8 in powers of x.

Original entry on oeis.org

1, -8, 28, -56, 62, 0, -148, 328, -419, 280, 140, -728, 1232, -1336, 848, 224, -1582, 2688, -3072, 2408, -742, -1568, 3836, -5264, 5306, -3744, 924, 2576, -5686, 7792, -8092, 6272, -2751, -1848, 6008, -9296, 10556, -9800, 6692, -2240, -3206, 8168, -11524
Offset: 8

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 - A001485, 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)^8 )); // 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, 8):
seq(a(n), n=8..50);  # Alois P. Heinz, Feb 07 2021
  • Mathematica
    nmax=50; CoefficientList[Series[(Product[(1 -(-x)^j), {j,nmax}] -1)^8, {x,0,nmax}], x]//Drop[#,8] & (* Ilya Gutkovskiy, Feb 07 2021 *)
Drop[CoefficientList[Series[(QPochhammer[-x] -1)^8, {x,0,102}], x], 8] (* G. C. Greubel, Sep 04 2023 *)
  • PARI
    my(N=55,x='x+O('x^N)); Vec((eta(-x)-1)^8) \\ Joerg Arndt, Sep 05 2023
  • SageMath
    from sage.modular.etaproducts import qexp_eta
m=100; k=8;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A001486_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A001486_list(m); a[k:] # G. C. Greubel, Sep 04 2023

Formula

a(n) = [x^n]( QPochhammer(-x) - 1 )^8. - 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.

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