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 1-5 of 5 results.

A160549 Omit first term from A160539 and divide by 7.

Original entry on oeis.org

0, 1, 5, 20, 70, 221, 646, 1772, 4614, 11490, 27537, 63808, 143514, 314279, 671872, 1405260, 2881030, 5799093, 11476452, 22357584, 42922558, 81284699, 151974124, 280739800, 512761178, 926568075, 1657448779, 2936506316, 5155349836, 8972488674, 15487146900
Offset: 0

Views

Author

N. J. A. Sloane, Nov 14 2009

Keywords

Comments

These are Watson's coefficients beta'_n on page 125.

Examples

			G.f. = x + 5*x^2 + 20*x^3 + 70*x^4 + 221*x^5 + 646*x^6 + ...

Links

Crossrefs

Cf. A160539.
Cf. Expansion of ((Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k) - 1)/k in powers of x: A014968 (k=2), A277968 (k=3), A277974 (k=5), this sequence (k=7), A277912 (k=11).

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[(Product[(1 - x^(7*j))/(1 - x^j)^7, {j, 1, nmax}] - 1)/7, {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *)
  • PARI
    x='x+O('x^66); concat([0],Vec(eta(x^7)/eta(x)^7-1)/7) \\ Joerg Arndt, Nov 27 2016

Formula

From Seiichi Manyama, Nov 07 2016: (Start)
a(n) = A160539(n)/7, n > 0.
G.f.: ((Product_{n>=1} (1 - x^(7*n))/(1 - x^n)^7) - 1)/7. (End)
a(n) ~ 2^(5/4) * exp(4*Pi*sqrt(2*n/7)) / (7^(13/4) * n^(9/4)). - Vaclav Kotesovec, Nov 10 2016

Extensions

Typo in definition corrected by Seiichi Manyama, Nov 07 2016

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

A274327 Expansion of Product_{n>=1} (1 - x^(4*n))/(1 - x^n)^4 in powers of x.

Original entry on oeis.org

1, 4, 14, 40, 104, 248, 560, 1200, 2474, 4924, 9520, 17928, 33008, 59528, 105408, 183536, 314744, 532208, 888382, 1465208, 2389808, 3857456, 6166096, 9766576, 15336816, 23888844, 36924656, 56659296, 86341664, 130710104, 196640576, 294059872, 437232746, 646561792
Offset: 0

Views

Author

Seiichi Manyama, Nov 07 2016

Keywords

Examples

			G.f.: 1 + 4*x + 14*x^2 + 40*x^3 + 104*x^4 + 248*x^5 + 560*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), this sequence (k=4), A277212 (k=5), A277283 (k=6), A160539 (k=7).
Cf. A083703.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 - x^(4*k))/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *)
(QPochhammer[x^4, x^4]/QPochhammer[x, x]^4 + O[x]^40)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)
  • PARI
    first(n)=my(x='x);Vec(prod(k=1,n,(1-x^(4*k))/(1-x^k)^4,1+O(x^(n+1)))) \\ Charles R Greathouse IV, Nov 07 2016

Formula

G.f.: Product_{n>=1} (1 - x^(4*n))/(1 - x^n)^4.
a(n) ~ 5*exp(Pi*sqrt(5*n/2)) / (2^(13/2) * n^(3/2)). - Vaclav Kotesovec, Nov 10 2016
G.f.: (x^4; x^4)inf/((x; x)_inf)^4, where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2016
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 1-5 of 5 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.