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.

A307462 Expansion of Product_{k>=1} (1+x^k)^((-1)^k*k^2).

Original entry on oeis.org

1, -1, 5, -14, 36, -97, 246, -593, 1423, -3351, 7699, -17432, 38901, -85545, 185862, -399220, 848080, -1783682, 3716584, -7675916, 15722127, -31951330, 64452707, -129102947, 256876062, -507854808, 997954125, -1949631802, 3787674152, -7319306458, 14071371173
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): A083365 (b=0), A284474 (b=1), this sequence (b=2).
Cf. A027998, A177155, A266964, A294749, A294750, A307460.

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 *)
nmax = 40; CoefficientList[Series[Product[(1 + x^(2*k))^(4*k^2) / (1 + x^(2*k - 1))^((2*k - 1)^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)))

Formula

a(n) ~ (-1)^n * exp(2*Pi*n^(3/4)/3 + 3*Zeta(3)/(4*Pi^2)) / (4*n^(5/8)). - Vaclav Kotesovec, Apr 09 2019

A284467 Expansion of Product_{k>=1} (1 + x^(2*k-1))^(2*k-1)/(1 + x^(2*k))^(2*k).

Original entry on oeis.org

1, 1, -2, 1, 2, -2, 0, -5, 10, 1, -15, 10, -1, 18, -39, 4, 50, -24, -14, -69, 165, -70, -83, -20, 154, 161, -550, 313, 55, 410, -960, 102, 1074, -406, -506, -1344, 3581, -1791, -833, -1833, 4995, 205, -6993, 2982, 2461, 7649, -19791, 9495, 4986, 9581, -26745, 0
Offset: 0

Views

Author

Seiichi Manyama, Apr 15 2017

Keywords

Links

Crossrefs

Cf. A224364, A262736, A278710, A281683, A284474.

Programs

  • Maple
    N:= 100: # to get a(0)..a(N)
P:= mul((1+x^(2*k-1))^(2*k-1)/(1+x^(2*k))^(2*k),k=1..N/2):
S:= series(P,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Apr 16 2017
  • Mathematica
    nmax = 60; CoefficientList[Series[Product[(1 + x^(2*k-1))^(2*k-1)/(1 + x^(2*k))^(2*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)

Formula

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 + x^k)^2)). - Ilya Gutkovskiy, Jun 20 2018

A295831 Expansion of Product_{k>=1} ((1 + x^(2*k))/(1 - x^(2*k-1)))^k.

Original entry on oeis.org

1, 1, 2, 4, 6, 11, 19, 30, 47, 76, 118, 181, 277, 417, 624, 929, 1367, 2001, 2913, 4210, 6056, 8665, 12328, 17466, 24640, 34600, 48395, 67442, 93625, 129520, 178588, 245429, 336252, 459324, 625613, 849762, 1151150, 1555378, 2096332, 2818630, 3780903, 5060240, 6757633, 9005106, 11975265
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 28 2017

Keywords

Crossrefs

Cf. A001935, A001936, A001937, A026007, A035528, A093160, A113415, A156616, A284474, A292037, A295832.

Programs

  • Mathematica
    nmax = 44; CoefficientList[Series[Product[((1 + x^(2 k))/(1 - x^(2 k - 1)))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 44; CoefficientList[Series[Exp[Sum[x^k (1 - (-1)^k x^k)/(k (1 - x^(2 k))^2), {k, 1, nmax}]], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} ((1 + x^(2*k))/(1 - x^(2*k-1)))^k.
G.f.: exp(Sum_{k>=1} x^k*(1 - (-1)^k*x^k)/(k*(1 - x^(2*k))^2)).
a(n) ~ exp(3*(7*Zeta(3))^(1/3) * n^(2/3) / 4 + Pi^2 * n^(1/3) / (12 * (7*Zeta(3))^(1/3)) - Pi^4 / (3024*Zeta(3)) - 1/24) * A^(1/2) * (7*Zeta(3))^(11/72) / (2^(11/8) * sqrt(3*Pi) * n^(47/72)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 28 2017

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

A295833 Expansion of e.g.f. Product_{k>=1} (1 + x^k)^((-1)^k/k).

Original entry on oeis.org

1, -1, 3, -11, 47, -279, 2089, -16057, 137409, -1417553, 15656651, -187422531, 2501688463, -34832785831, 529520417217, -8723102543009, 146573712239489, -2670058109819937, 52017332039568019, -1041334898093864443, 22335551258991482991, -502509800119879530551, 11641825391540821682393
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 28 2017

Keywords

Examples

			E.g.f.: Sum_{n>=0} a(n)*x^n/n! = ((1 + x^2)^(1/2)*(1 + x^4)^(1/4)*(1 + x^6)^(1/6)* ...)/((1 + x)*(1 + x^3)^(1/3)*(1 + x^5)^(1/5)* ...) = 1 - x + 3*x^2/2! - 11*x^3/3! + 47*x^4/4! - 279*x^5/5! + 2089*x^6/6! - 16057*x^7/7! + ...

Crossrefs

Cf. A028342, A168243, A206303, A284474, A294356, A295792, A295834.

Programs

  • Maple
    a:=series(mul((1+x^k)^((-1)^k/k),k=1..100),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[Product[(1 + x^k)^((-1)^k/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Showing 1-5 of 5 results.

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