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.

A321438 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^n.

Original entry on oeis.org

1, 3, 28, 239, 3126, 45990, 823544, 16711423, 387440173, 9990235398, 285311670612, 8913939907598, 302875106592254, 11111328602501550, 437893920912786408, 18446462594437808127, 827240261886336764178, 39346258082220810086373, 1978419655660313589123980, 104857499999905732078938574
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 09 2018

Keywords

Links

Crossrefs

Cf. A000593, A023887, A078306, A078307, A186633, A284900, A284926, A284927, A321385.

Programs

  • Magma
    m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[(k*x)^k/(1+(k*x)^k): k in [1..m]]) ));  // G. C. Greubel, Nov 11 2018
  • Mathematica
    Table[Sum[(-1)^(n/d + 1) d^n, {d, Divisors[n]}], {n, 20}]
nmax = 20; Rest[CoefficientList[Series[Sum[(k x)^k/(1 + (k x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 20; Rest[CoefficientList[Series[Log[Product[(1 + k^k x^k)^(1/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
  • PARI
    a(n) = sumdiv(n, d, (-1)^(n/d+1)*d^n); \\ Michel Marcus, Nov 09 2018

Formula

G.f.: Sum_{k>=1} (k*x)^k/(1 + (k*x)^k).
L.g.f.: log(Product_{k>=1} (1 + k^k*x^k)^(1/k)) = Sum_{n>=1} a(n)*x^n/n.
a(n) ~ n^n. - Vaclav Kotesovec, Nov 10 2018

A321387 Expansion of Product_{k>=1} (1 + x^k)^(k^(k-1)).

Original entry on oeis.org

1, 1, 2, 11, 74, 708, 8583, 127424, 2239965, 45514345, 1049365071, 27061132159, 771695223819, 24109698083919, 818914886275467, 30044684789498522, 1184048086192376822, 49883929845112421452, 2237287911899357657492, 106426388125032988691636, 5352033610656721914626572
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 08 2018

Keywords

Comments

Weigh transform of A000169.

Links

Crossrefs

Cf. A000169, A023879, A261053, A283335, A321385, A321388.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1+x^k)^(k^(k-1)): k in [1..m]]) )); // G. C. Greubel, Nov 09 2018
  • Maple
    a:=series(mul((1+x^k)^(k^(k-1)),k=1..100),x=0,21): seq(coeff(a,x,n),n=0..20); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 20; CoefficientList[Series[Product[(1 + x^k)^(k^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^d, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 20}]
  • PARI
    seq(n)={Vec(exp(sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d^d ) * x^k/k) + O(x*x^n)))} \\ Andrew Howroyd, Nov 09 2018
  • PARI
    m=30; x='x+O('x^m); Vec(prod(k=1,m,(1+x^k)^(k^(k-1)))) \\ G. C. Greubel, Nov 09 2018

Formula

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d^d ) * x^k/k).
a(n) ~ n^(n-1) * (1 + exp(-1)/n + (3*exp(-1)/2 + 2*exp(-2))/n^2). - Vaclav Kotesovec, Nov 09 2018

A076723 Sum_{d divides n} (-d)^d.

Original entry on oeis.org

-1, 3, -28, 259, -3126, 46632, -823544, 16777475, -387420517, 9999996878, -285311670612, 8916100495144, -302875106592254, 11112006824734476, -437893890380862528, 18446744073726329091, -827240261886336764178, 39346408075296150201567
Offset: 1

Views

Author

Vladeta Jovovic, Oct 27 2002

Keywords

Links

Crossrefs

Cf. A062796, A321385.

Programs

  • PARI
    a(n) = sumdiv(n, d, (-d)^d); \\ Michel Marcus, Dec 22 2018

Formula

G.f.: Sum_{n>0} (-n*x)^n/(1-x^n).
Showing 1-3 of 3 results.

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