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.

A277212 Expansion of Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^5 in powers of x.

Original entry on oeis.org

1, 5, 20, 65, 190, 505, 1260, 2970, 6700, 14535, 30520, 62235, 123720, 240340, 457380, 854190, 1568230, 2834120, 5048140, 8871450, 15396690, 26410860, 44811440, 75254240, 125162100, 206275505, 337032360, 546183425, 878270360, 1401857550, 2221862260
Offset: 0

Views

Author

Seiichi Manyama, Nov 07 2016

Keywords

Comments

In general, for fixed m > 1, if g.f. = Product_{k>=1} (1 - x^(m*k))/(1 - x^k)^m, then a(n, m) ~ exp(Pi*sqrt(2*n*(m-1/m)/3)) * (m^2 - 1)^(m/4) / (2^(3*m/4 + 1/2) * 3^(m/4) * m^(m/4 + 1/2) * n^(m/4 + 1/2)). - Vaclav Kotesovec, Nov 10 2016

Examples

			G.f.: 1 + 5*x + 20*x^2 + 65*x^3 + 190*x^4 + 505*x^5 + 1260*x^6 + ...

Links

Crossrefs

Cf. Expansion of Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k in powers of x: A015128 (k=2), A273845 (k=3), A274327 (k=4), this sequence (k=5), A160539 (k=7).
Cf. A109064.

Programs

  • Maple
    N:= 100: # to get a(0)..a(N)
S:= series(mul((1-x^(5*n))/(1-x^n)^5,n=1..N),x,N+1):
seq(coeff(S,x,n),n=0..N); # Robert Israel, Nov 09 2016
  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))/(1 - x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *)
(QPochhammer[x^5, x^5]/QPochhammer[x, x]^5 + O[x]^40)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)
  • PARI
    first(n)=my(x='x); Vec(prod(k=1, n, (1-x^(5*k))/(1-x^k)^5, 1+O(x^(n+1)))) \\ Charles R Greathouse IV, Nov 07 2016
  • PARI
    x='x+O('x^66); Vec(eta(x^5)/eta(x)^5) \\ Joerg Arndt, Nov 27 2016

Formula

G.f.: Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^5.
a(n) ~ exp(4*Pi*sqrt(n/5)) / (sqrt(2) * 5^(7/4) * n^(7/4)). - Vaclav Kotesovec, Nov 10 2016
G.f.: (x^5; x^5)inf/((x; x)_inf)^5, where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2016
a(0) = 1, a(n) = (5/n)*Sum_{k=1..n} A285896(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017

A273845 Expansion of Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^3 in powers of x.

Original entry on oeis.org

1, 3, 9, 21, 48, 99, 198, 375, 693, 1236, 2160, 3681, 6168, 10140, 16434, 26235, 41376, 64449, 99342, 151530, 229032, 343068, 509760, 751509, 1099998, 1598925, 2309274, 3314541, 4729920, 6711993, 9474624, 13306506, 18598437, 25874460, 35838288, 49427640, 67892592
Offset: 0

Views

Author

Seiichi Manyama, Nov 07 2016

Keywords

Examples

			G.f.: 1 + 3*x + 9*x^2 + 21*x^3 + 48*x^4 + 99*x^5 + 198*x^6 + ...

Links

Crossrefs

Expansion of Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k in powers of x: A015128 (k=2), this sequence (k=3), A274327 (k=4), A277212 (k=5), A277283 (k=6), A160539 (k=7).
Cf. A005928, A132972.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 - x^(3*k))/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *)
(QPochhammer[x^3, x^3]/QPochhammer[x, x]^3 + O[x]^40)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)
  • PARI
    first(n)=my(x='x); Vec(prod(k=1, n, (1-x^(3*k))/(1-x^k)^3, 1+O(x^(n+1)))) \\ Charles R Greathouse IV, Nov 07 2016
  • PARI
    lista(nn) = {q='q+O('q^nn); Vec(eta(q^3)/eta(q)^3)} \\ Altug Alkan, Mar 20 2018

Formula

G.f.: Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^3.
a(n) ~ exp(4*Pi*sqrt(n)/3) / (9*sqrt(2)*n^(5/4)). - Vaclav Kotesovec, Nov 10 2016
G.f.: (x^3; x^3)inf/((x; x)_inf)^3, where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2016
a(0) = 1, a(n) = (3/n)*Sum_{k=1..n} A078708(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017
It appears that the g.f. A(x) = F(x)^3, where F(x) = exp( Sum_{n >= 0} x^(3*n+1)/((3*n + 1)*(1 - x^(3*n+1))) + x^(3*n+2)/((3*n + 2)*(1 - x^(3*n + 2))) ). Cf. A132972. - Peter Bala, Dec 23 2021

A083703 Expansion of eta(q)^4/eta(q^4) in powers of q.

Original entry on oeis.org

1, -4, 2, 8, -4, -8, -8, 16, 6, -12, 8, 8, -8, -24, 0, 16, 12, -16, 10, 24, -8, -16, -24, 16, 8, -28, 8, 32, -16, -8, 0, 32, 6, -32, 16, 16, -12, -40, -24, 16, 24, -16, 16, 40, -8, -40, 0, 32, 24, -36, 10, 16, -24, -24, -32, 48, 0, -32, 24, 24, -16, -40, 0, 48, 12, -16, 16, 56, -16, -32, -48, 16, 30, -64, 8, 40, -24
Offset: 0

Views

Author

Michael Somos, May 04 2003

Keywords

Comments

Euler transform of period 4 sequence [ -4,-4,-4,-3,...].

Links

Crossrefs

A080965(n)=(-1)^n a(n). a(2n)=0 iff n in A004215 (checked up to n=343).
a(2n)=0 iff A005875(n)=0.
Cf. A274327, A285895.

Programs

  • Maple
    with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d, 4)=0, -3, -4), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Jan 07 2017
  • Mathematica
    CoefficientList[QPochhammer[x]^4/QPochhammer[x^4] + O[x]^80, x] (* Jean-François Alcover, Sep 19 2016 *)
  • PARI
    a(n)=if(n<0,0,X=x+x*O(x^n); polcoeff(eta(X)^4/eta(X^4),n))

Formula

G.f.: Product_{n>0} (1-x^n)^4/(1-x^(4n)).
a(0) = 1, a(n) = -(4/n)*Sum_{k=1..n} A285895(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017

A277283 Expansion of Product_{n>=1} (1 - x^(6*n))/(1 - x^n)^6 in powers of x.

Original entry on oeis.org

1, 6, 27, 98, 315, 918, 2491, 6366, 15498, 36182, 81501, 177876, 377558, 781626, 1582173, 3137832, 6108051, 11687598, 22012816, 40855674, 74799828, 135210868, 241511115, 426570624, 745516240, 1290006276, 2211202692, 3756468658, 6327617862, 10572763842
Offset: 0

Views

Author

Seiichi Manyama, Nov 07 2016

Keywords

Examples

			G.f.: 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 918*x^5 + 2491*x^6 + ...

Links

Crossrefs

Cf. Expansion of Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k in powers of x: A015128 (k=2), A273845 (k=3), A274327 (k=4), A277212 (k=5), this sequence (k=6), A160539 (k=7).

Programs

  • Mathematica
    (QPochhammer[x^6, x^6]/QPochhammer[x, x]^6 + O[x]^40)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^(6*k))/(1 - x^k)^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 21 2016 *)
  • PARI
    first(n)=my(x='x); Vec(prod(k=1, n, (1-x^(6*k))/(1-x^k)^6, 1+O(x^(n+1)))) \\ Charles R Greathouse IV, Nov 07 2016

Formula

G.f.: Product_{n>=1} (1 - x^(6*n))/(1 - x^n)^6.
G.f.: (x^6; x^6)inf/((x; x)_inf)^6, where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2016
a(n) ~ 35*sqrt(35) * exp(sqrt(35*n)*Pi/3) / (3456*sqrt(3)*n^2). - Vaclav Kotesovec, Nov 21 2016
Showing 1-4 of 4 results.

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