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.

A247003 Decimal expansion of a constant related to A034691.

Original entry on oeis.org

1, 3, 9, 7, 6, 4, 9, 0, 0, 5, 0, 8, 3, 6, 5, 0, 2, 8, 5, 0, 6, 5, 0, 7, 4, 5, 9, 8, 5, 2, 6, 7, 9, 1, 1, 5, 9, 0, 0, 7, 8, 1, 1, 4, 2, 9, 4, 4, 0, 7, 2, 8, 9, 9, 6, 4, 8, 3, 8, 7, 4, 0, 4, 8, 8, 5, 4, 6, 6, 5, 7, 2, 0, 6, 6, 0, 8, 3, 3, 8, 5, 7, 8, 2, 0, 7, 5, 7, 3, 3, 2, 3, 3, 1, 0, 2, 4, 8, 2, 0, 4, 0, 0, 1, 5
Offset: 1

Views

Author

Vaclav Kotesovec, Sep 09 2014

Keywords

Examples

			1.397649005083650285065074598526791159007811429440728996483874...

Links

Crossrefs

Cf. A034691, A034899, A261043, A304962, A319918.

Programs

  • Maple
    evalf(exp(sum(1/(k*(2^k-2)), k=2..infinity)), 100)

Formula

Equals exp( Sum_{k>=2} 1/(k*(2^k-2)) ).

A319919 Expansion of Product_{k>=1} (1 + x^k)^(2^k-1).

Original entry on oeis.org

1, 1, 3, 10, 25, 70, 182, 476, 1220, 3122, 7883, 19794, 49340, 122237, 301114, 737923, 1799597, 4369204, 10563800, 25441377, 61048713, 145988775, 347981713, 826921992, 1959363778, 4629903905, 10911757432, 25652950459, 60165831361, 140792215037, 328750398275, 766041930160, 1781452975346
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 01 2018

Keywords

Comments

Convolution of A081362 and A102866.
Weigh transform of A000225.

Links

Crossrefs

Cf. A000225, A007070, A081362, A098407, A102866, A319918.

Programs

  • Maple
    a:=series(mul((1+x^k)^(2^k-1),k=1..100),x=0,33): seq(coeff(a,x,n),n=0..32); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 32; CoefficientList[Series[Product[(1 + x^k)^(2^k - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[Exp[Sum[(-1)^(k + 1) x^k/(k (1 - x^k) (1 - 2 x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (2^d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]

Formula

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)*(1 - 2*x^k))).
a(n) ~ c * exp(2*sqrt(n) - 1/2) * 2^(n-1) / (A079555 * sqrt(Pi) * n^(3/4)), where c = exp(Sum_{k>=2} (-1)^(k-1)/(k*(2^(k-1)-1))) = 0.6602994483152065685... - Vaclav Kotesovec, Sep 15 2021

A347011 Euler transform of j-> ceiling(2^(j-2)).

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 43, 93, 207, 453, 999, 2185, 4796, 10470, 22871, 49815, 108427, 235515, 511074, 1107248, 2396299, 5179169, 11181877, 24113939, 51949572, 111801422, 240381703, 516355235, 1108186951, 2376314763, 5091422730, 10900063776, 23317805916
Offset: 0

Views

Author

Alois P. Heinz, Aug 10 2021

Keywords

Comments

Differs from A206301 first at n=10.

Links

Crossrefs

Cf. A034691, A034899, A166861, A187251, A206301, A319918.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      b(n-i*j, i-1)*binomial(ceil(2^(i-2))+j-1, j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..35);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(d*
       ceil(2^(d-2)), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..35);
  • Mathematica
    CoefficientList[Series[1/(1-x) * Product[1/(1 - x^k)^(2^(k-2)), {k, 2, 40}], {x, 0, 40}], x] (* Vaclav Kotesovec, Aug 11 2021 *)

Formula

G.f.: Product_{j>0} 1/(1-x^j)^ceiling(2^(j-2)).

A308447 Expansion of Sum_{k>=1} mu(k)*log(1 + x^k/((1 - x^k)*(1 - 2*x^k)))/k.

Original entry on oeis.org

1, 2, 4, 5, 8, 8, 16, 25, 52, 98, 192, 345, 640, 1162, 2164, 4050, 7680, 14534, 27648, 52479, 99956, 190554, 364544, 698525, 1341848, 2580790, 4971616, 9587565, 18513920, 35790276, 69271552, 134211600, 260297012, 505286430, 981714296, 1908881520, 3714580480, 7233615306
Offset: 1

Views

Author

Ilya Gutkovskiy, May 27 2019

Keywords

Comments

Inverse Euler transform of A000225.

Links

Crossrefs

Cf. A000225, A001037, A008683, A059966, A319918.

Programs

  • Mathematica
    nmax = 38; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + x^k/((1 - x^k) (1 - 2 x^k))]/k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 50; s = ConstantArray[0, nmax]; Do[s[[j]] = j*(2^ j - 1) - Sum[s[[d]]*(2^(j - d) - 1), {d, 1, j - 1}], {j, 1, nmax}]; Table[Sum[MoebiusMu[k/d]*s[[d]], {d, Divisors[k]}]/k, {k, 1, nmax}] (* Vaclav Kotesovec, Aug 10 2019 *)

Formula

-1 + Product_{n>=1} 1/(1 - x^n)^a(n) = g.f. of A000225.
a(n) ~ 2^n/n. - Vaclav Kotesovec, May 28 2019
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.