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.

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A294461 E.g.f.: exp(-Sum_{n>=1} A050999(n) * x^n).

Original entry on oeis.org

1, -1, -1, -55, 217, -2441, 41911, -343519, 10531025, -123024817, 2722259791, -64395229031, 1218005521129, -36874422541945, 785879799954887, -25331247487596751, 708096286059632161, -21422225147712360929, 741754828422824400415
Offset: 0

Views

Author

Seiichi Manyama, Oct 31 2017

Keywords

Links

Crossrefs

E.g.f.: exp(-Sum_{n>=1} (Sum_{d|n and d is odd} d^k) * x^n): A294459 (k=0), A294460 (k=1), this sequence (k=2).
Cf. A050999, A285069, A294395.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp(-sum(k=1, N, sumdiv(k, d, d^2*(d%2))*x^k))))

Formula

a(0) = 1 and a(n) = (-1) * (n-1)! * Sum_{k=1..n} k*A050999(k)*a(n-k)/(n-k)! for n > 0.

A294458 E.g.f.: Product_{n>=1} (1 - x^(2*n-1))^(1/(2*n-1)).

Original entry on oeis.org

1, -1, 0, -2, 8, -24, 64, -160, 8448, -86912, 509696, -1449216, 44615680, -366395392, 3315376128, -190488356864, 4591008579584, -33244620718080, 86342088982528, -2543409132470272, 136456182420996096, -5644134983026343936, 103753337226615848960
Offset: 0

Views

Author

Seiichi Manyama, Oct 31 2017

Keywords

Links

Crossrefs

Cf. A081362, A206303, A285069, A294459.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp(-sum(k=1, N, sumdiv(k, d, d%2)*x^k/k))))

Formula

E.g.f.: exp(-Sum_{n>=1} A001227(n) * x^n / n).
a(0) = 1 and a(n) = (-1) * (n-1)! * Sum_{k=1..n} A001227(k)*a(n-k)/(n-k)! for n > 0.

A319669 Expansion of Product_{k>=1} (1 - x^k)^(2*k-1).

Original entry on oeis.org

1, -1, -3, -2, 1, 10, 13, 15, -1, -30, -63, -89, -80, -14, 131, 304, 493, 561, 434, -32, -836, -1895, -2960, -3583, -3240, -1338, 2401, 8004, 14499, 20494, 23369, 20401, 8567, -13741, -46408, -85717, -124027, -149612, -147167, -101002, 2520, 168026, 388077, 634914
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 25 2018

Keywords

Crossrefs

Cf. A000203, A001157, A073592, A253289, A255835, A276551, A278945, A285069.

Programs

  • Maple
    a:=series(mul((1-x^k)^(2*k-1),k=1..100),x=0,44): seq(coeff(a,x,n),n=0..43); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 43; CoefficientList[Series[Product[(1 - x^k)^(2 k - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 43; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, k] - 2 DivisorSigma[2, k]) x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (1 - 2 d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 43}]

Formula

G.f.: exp(Sum_{k>=1} (sigma_1(k) - 2*sigma_2(k))*x^k/k).
Previous Showing 11-13 of 13 results.

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