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.

A177155 G.f.: exp( Integral (theta_3(x)^8-1)/(16x) dx ), where theta_3(x) = 1 + Sum_{n>=1} 2*x^(n^2) is a Jacobi theta function.

Original entry on oeis.org

1, 1, 4, 13, 35, 87, 217, 539, 1291, 2999, 6880, 15595, 34738, 76202, 165282, 354655, 752546, 1580514, 3289337, 6787085, 13887937, 28195434, 56824772, 113729640, 226104615, 446665922, 877063515, 1712252521, 3324245063, 6419561961
Offset: 0

Views

Author

Paul D. Hanna, May 03 2010, May 08 2010

Keywords

Comments

Compare to g.f. of partitions in which no parts are multiples of 4:
g.f. of A001935 = exp( Integral (theta_3(x)^4-1)/(8x) dx ).

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 13*x^3 + 35*x^4 + 87*x^5 +...
log(A(x)) = x + 7*x^2/2 + 28*x^3/3 + 71*x^4/4 + 126*x^5/5 +...+ A008457(n)*x^n/n +...

Links

Crossrefs

Cf. A138503, A008457, A001935, A000143, A307462.

Programs

  • Mathematica
    nmax = 40; Abs[CoefficientList[Series[Product[1/(1 - x^k)^((-1)^k*k^2), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Apr 10 2019 *)
nmax = 40; CoefficientList[Series[Product[(1 + x^(2*k - 1))^((2*k - 1)^2)/(1 - x^(2*k))^(4*k^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 10 2019 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n, sumdiv(m,d,(-1)^(m-d)*d^3)*x^m/m)+x*O(x^n)),n)}
  • PARI
    {a(n)=local(theta3=1+sum(m=1,sqrtint(2*n+2),2*x^(m^2)+x*O(x^n)));polcoeff(exp(intformal((theta3^8-1)/(16*x))),n)}

Formula

G.f.: exp( Sum_{n>=1} A008457(n)*x^n/n ) where A008457(n) = Sum_{d|n} (-1)^(n-d)*d^3.
a(n) ~ exp(2*Pi*n^(3/4)/3 - Zeta(3)/Pi^2) / (4*n^(5/8)). - Vaclav Kotesovec, Apr 10 2019

A307460 Expansion of Product_{k>=1} (1-x^k)^((-1)^k*k^2).

Original entry on oeis.org

1, 1, -3, 6, -4, -15, 54, -87, 63, 79, -405, 912, -1363, 1193, 510, -4900, 12512, -21582, 26512, -16540, -24585, 113682, -255045, 419931, -519210, 377176, 267957, -1703694, 4090424, -7179222, 9895981, -9897664, 3337614, 14790666, -49171217, 100903743
Offset: 0

Views

Author

Seiichi Manyama, Apr 09 2019

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = (-1)^(n+1) * n^2, g(n) = 1.

Links

Crossrefs

Product_{k>=1} (1-x^k)^((-1)^k*k^b): A010054 (b=0), A281781 (b=1), this sequence (b=2).
Cf. A266964, A283263, A307462.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[(1 - x^k)^((-1)^k*k^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 09 2019 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-x^k)^((-1)^k*k^2)))

A307497 Expansion of Product_{k>=1} (1+x^k)^((-1)^k*k^k).

Original entry on oeis.org

1, -1, 5, -32, 294, -3527, 51589, -894706, 17978610, -410803143, 10517824035, -298204099693, 9273022031794, -313755862498513, 11474175971184267, -450960476552715192, 18954545423649435646, -848383466771831169101, 40285210722052785437974
Offset: 0

Views

Author

Seiichi Manyama, Apr 10 2019

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = (-1)^(n+1) * n^n, g(n) = -1.

Links

Crossrefs

Cf. A083365, A261053, A266964, A284474, A307462.

Programs

  • Mathematica
    nmax=20; CoefficientList[Series[Product[(1+x^k)^((-1)^k*k^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 12 2019 *)
  • PARI
    N=20; x='x+O('x^N); Vec(prod(k=1, N, (1+x^k)^((-1)^k*k^k)))

Formula

a(n) ~ (-1)^n * n^n * (1 + exp(-1)/n + (exp(-1)/2 + 5*exp(-2))/n^2). - Vaclav Kotesovec, Apr 12 2019

A307484 Expansion of Product_{k>=1} 1/(1+x^k)^((-1)^k*k^2).

Original entry on oeis.org

1, 1, -4, 5, 3, -17, 33, -61, 67, 63, -392, 803, -1070, 898, 482, -4449, 11362, -18630, 21105, -11067, -24871, 103562, -227004, 359040, -417697, 266106, 312987, -1578543, 3635615, -6157911, 8155892, -7689028, 1502546, 14707881, -44539735, 87849728, -136927058, 171008704
Offset: 0

Views

Author

Seiichi Manyama, Apr 10 2019

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = (-1)^n * n^2, g(n) = -1.

Links

Crossrefs

Product_{k>=1} 1/(1+x^k)^((-1)^k*k^b): A029838 (b=0), A284467 (b=1), this sequence (b=2).
Cf. A177155, A307462.

Programs

  • Mathematica
    m = 37; CoefficientList[Series[Product[1/(1+x^k)^((-1)^k*k^2), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 14 2021 *)
nmax = 40; CoefficientList[Series[Product[(1 + x^(2*k - 1))^((2*k - 1)^2)/(1 + x^(2*k))^(4*k^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 14 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1+x^k)^((-1)^k*k^2)))
Showing 1-4 of 4 results.

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