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 1-3 of 3 results.

A318118 a(n) = [x^n] exp(Sum_{k>=1} x^k*(1 + (n - 3)*x^k)/(k*(1 - x^k)^3)).

Original entry on oeis.org

1, 1, 3, 10, 40, 150, 616, 2456, 10102, 41400, 171526, 712111, 2972115, 12434993, 52195414, 219567909, 925704792, 3909841659, 16541598215, 70085877919, 297347922785, 1263046810334, 5370930049915, 22861883482838, 97402827429118, 415332438952517, 1772380322197432
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 18 2018

Keywords

Comments

For n > 2, a(n) is the n-th term of the Euler transform of n-gonal numbers.

Links

Crossrefs

Cf. A000294, A023871, A274998, A278767, A278768, A278769.

Programs

  • Mathematica
    Table[SeriesCoefficient[Exp[Sum[x^k (1 + (n - 3) x^k)/(k (1 - x^k)^3), {k, 1, n}]], {x, 0, n}], {n, 0, 26}]

Formula

a(n) ~ c * d^n / sqrt(n), where d = 4.3505530790182509701639869563721679988879373943131559534408716195123... and c = 0.2276354216252041005336767937139336687746108521151301186102034... - Vaclav Kotesovec, Aug 18 2018

A317019 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*binomial(k+2,3)).

Original entry on oeis.org

1, 1, 9, 39, 155, 570, 2131, 7599, 26667, 90996, 305144, 1004173, 3254123, 10385884, 32704819, 101678860, 312435675, 949498206, 2855953018, 8507079361, 25108844890, 73468004480, 213201630328, 613871526178, 1754365814430, 4978113020152, 14029639217532, 39281646364737
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 19 2018

Keywords

Comments

Euler transform of A002417.

Links

Crossrefs

Cf. A000391, A002417, A278767, A279217, A305653, A317017, A317020, A317021.

Programs

  • Maple
    a:=series(mul(1/(1-x^k)^(k*binomial(k+2,3)),k=1..100),x=0,28): seq(coeff(a,x,n),n=0..27); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 27; CoefficientList[Series[Product[1/(1 - x^k)^(k Binomial[k + 2, 3]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 27; CoefficientList[Series[Exp[Sum[x^k (1 + 3 x^k)/(k (1 - x^k)^5), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^3 (d + 1) (d + 2)/6, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 27}]

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^A002417(k).
G.f.: exp(Sum_{k>=1} x^k*(1 + 3*x^k)/(k*(1 - x^k)^5)).
a(n) ~ 1/(2^(601/720) * 3^(359/480) * 7^(119/1440) * n^(839/1440) * Pi^(1/240)) * exp(-Zeta(3)/(12 * Pi^2) + (491 * Zeta(5))/(400 * Pi^4) - (2250423 * Zeta(5)^3)/(10 * Pi^14) + (103355177121 * Zeta(5)^5)/(10 * Pi^24) + Zeta'(-3)/2 + ((-7 * 7^(1/6) * Pi)/(1200 * 2^(1/3) * sqrt(3)) + (27783 * sqrt(3) * 7^(1/6) * Zeta(5)^2)/(40 * 2^(1/3) * Pi^9) - (614365479 * sqrt(3) * 7^(1/6) * Zeta(5)^4)/(16 * 2^(1/3) * Pi^19)) * n^(1/6) + ((-63 * 7^(1/3) * Zeta(5))/(10 * 2^(2/3) * Pi^4) + (214326 * 14^(1/3) * Zeta(5)^3)/Pi^14) * n^(1/3) + ((sqrt(7/3) * Pi)/30 - (1701 * sqrt(21) * Zeta(5)^2)/(2 * Pi^9)) * sqrt(n) + ((27 * 7^(2/3) * Zeta(5))/(2 * 2^(1/3) * Pi^4)) * n^(2/3) + ((2 * 2^(1/3) * sqrt(3) * Pi)/(5 * 7^(1/6))) * n^(5/6)). - Vaclav Kotesovec, Jul 28 2018

A295828 Expansion of Product_{k>=1} 1/(1 - x^k)^(2*k*(2*k-1)).

Original entry on oeis.org

1, 2, 15, 58, 235, 862, 3122, 10664, 35639, 115164, 363806, 1122050, 3393316, 10068006, 29374056, 84347944, 238713339, 666419456, 1836986443, 5003473866, 13476019215, 35912177618, 94746481999, 247597696802, 641205816641, 1646268490598, 4192059724668, 10590937903412, 26556243826240
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 28 2017

Keywords

Comments

Euler transform of A002939.

Links

  • M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
  • M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
  • N. J. A. Sloane, Transforms

Crossrefs

Cf. A002939, A161870, A253289, A258347, A258348, A278767.

Programs

  • Mathematica
    nmax = 28; CoefficientList[Series[Product[1/(1 - x^k)^(2 k (2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[2 d^2 (2 d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 28}]

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^A002939(k).
a(n) ~ exp(2^(5/2) * Pi * n^(3/4) / (3^(5/4) * 5^(1/4)) - Zeta(3) * sqrt(15*n) / Pi^2 - 15^(5/4) * Zeta(3)^2 * n^(1/4) / (2^(3/2) * Pi^5) - Zeta(3) / Pi^2 - 75*Zeta(3)^3 / (2*Pi^8) - 1/6) * A^2 / (2^(4/3) * 15^(1/12) * Pi^(1/6) * n^(7/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 28 2017
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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