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.

Previous Showing 11-17 of 17 results.

A158947 Inverse Euler transform of A156217.

Original entry on oeis.org

1, 4, 21, 598, 1555, 497652, 299593, 320361028, 1178277701, 357046721884, 67546215517, 19351522090518670, 61054982558011, 1502551437369035044, 33657152197069739919, 45463945109198918808616, 128583032925805678351
Offset: 1

Views

Author

Vladeta Jovovic, Mar 31 2009

Keywords

Links

Crossrefs

Cf. A008683, A156217, A217872.

Programs

  • Maple
    A158947 := proc(n) add(numtheory[sigma](d)^d*numtheory[mobius](n/d),d=numtheory[divisors](n))/n ; end: seq( A158947(n),n=1..40) ; # R. J. Mathar, Apr 02 2009
# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(A156217):
seq(a(n), n = 1..17); # Peter Luschny, Nov 21 2022
  • Mathematica
    f[n_] := Block[{d = Divisors@n}, Plus @@ (DivisorSigma[1, d]^d*MoebiusMu[n/d])/n]; Array[f, 17] (* Robert G. Wilson v, May 04 2009 *)

Formula

a(n) = (1/n)*Sum_{d|n} sigma(d)^d*moebius(n/d).

Extensions

Extended by R. J. Mathar, Apr 02 2009

A305754 Inverse Euler transform of n^n.

Original entry on oeis.org

1, 3, 23, 223, 2800, 42576, 763220, 15734388, 366715248, 9533817400, 273549419552, 8586984241870, 292755986184548, 10772849583399474, 425587711650564816, 17966217346985801150, 807152054953801845760, 38451365602113352159320, 1936082850634342992601636
Offset: 1

Views

Author

Seiichi Manyama, Jun 10 2018

Keywords

Examples

			(1-x)^(-1) * (1-x^2)^(-3) * (1-x^3)^(-23) * (1-x^4)^(-223) * ... = 1 + x + 4*x^2 + 27*x^3 + 256*x^4 + ... .

Links

Crossrefs

Cf. A000312, A112354, A305787.

Programs

  • Maple
    # The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(n -> n^n):
seq(a(n), n = 1..19); # Peter Luschny, Nov 21 2022
  • Mathematica
    n = 20; s = {};
For[i = 1, i <= n, i++, AppendTo[s, i*i^i - Sum[s[[d]]*(i-d)^(i-d), {d, i - 1}]]];
Table[Sum[If[Divisible[i, d], MoebiusMu[i/d], 0]*s[[d]], {d, 1, i}]/i, {i, n}] (* Jean-François Alcover, May 10 2019 *)

Formula

Product_{k>=1} 1/(1-x^k)^{a(k)} = Sum_{n>=0} (n * x)^n.
a(n) ~ n^n. - Vaclav Kotesovec, Oct 09 2019

A316149 Inverse Euler transform of Thue-Morse sequence A001285.

Original entry on oeis.org

2, -1, -1, 2, -3, 3, 0, -4, 6, -6, 6, -1, -12, 24, -29, 23, 9, -64, 114, -132, 81, 78, -333, 577, -627, 279, 610, -1896, 2979, -2911, 672, 4232, -10754, 15576, -13515, -591, 28098, -61548, 81664, -60408, -27030, 180784, -351081, 425892, -253838, -281760, 1140396, -1995767, 2195952, -930876
Offset: 1

Views

Author

Seiichi Manyama, Jun 25 2018

Keywords

Examples

			(1-x)^(-2)*(1-x^2)*(1-x^3)*(1-x^4)^(-2)* ... = 1 + 2*x + 2*x^2 + x^3 + 2*x^4 + ... .

Links

Crossrefs

Cf. A001285, A029878.

Programs

  • Maple
    # The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(A001285):
seq(a(n), n = 1..50); # Peter Luschny, Nov 21 2022

Formula

Product_{k>=1} (1-x^k)^(-a(k)) = 1 + Sum_{k>=1} A001285(k)*x^k.

A018243 Inverse Euler transform of A000931.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 7, 8, 11, 13, 17, 21, 28, 34, 45, 56, 73, 92, 120, 151, 197, 250, 324, 414, 537, 687, 892, 1145, 1484, 1911, 2479, 3196, 4148, 5359, 6954, 9000, 11687, 15140, 19672, 25516, 33166, 43065, 56010, 72784, 94716, 123185, 160380, 208740, 271913, 354123, 461529, 601436, 784209, 1022505, 1333856
Offset: 1

Views

Author

N. J. A. Sloane, David Broadhurst

Keywords

Examples

			x^3 + x^5 + x^7 + x^8 + x^9 + x^10 + 2*x^11 + 2*x^12 + 3*x^13 + 3*x^14 + ...

Links

Crossrefs

Cf. A000931, A113788.

Programs

  • Maple
    # The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(A000931):
seq(a(n), n = 1..65); # Peter Luschny, Nov 21 2022
  • Mathematica
    a[n_] := (1/n)*Sum[ MoebiusMu[n/d]*Floor[ Re[ N[ RootSum[ -1-#+#^3&, #^d& ]]]] , {d, Divisors[n]}]; a[2]=0; Table[a[n], {n, 1, 65}] (* Jean-François Alcover, Oct 05 2012, after Michael Somos *)
  • Sage
    z = PowerSeriesRing(ZZ, 'z').gen().O(30)
r = (1 - (z**2 + z**3))/(1 - z**2)
F = -z*r.derivative()/r
[sum(moebius(n//d)*F[d] for d in divisors(n))//n for n in range(1, 24)] # F. Chapoton, Apr 25 2020

Formula

a(n) = A113788(n) unless n=2. - Michael Somos, Apr 06 2012
Reciprocal of g.f. of A000931 = (1 - x^2 - x^3) / (1 - x^2) = 1 - x^3 - x^5 - x^7 - x^9 - ... = Product_{k>0} (1 - x^k)^a(n). - Michael Somos, Jul 17 2012
a(n) ~ A060006^n / n. - Vaclav Kotesovec, Oct 09 2019

Extensions

More terms from Joerg Arndt, Jul 18 2012

A057772 Inverse Euler transform of A000016.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 4, 4, 12, 15, 34, 55, 110, 190, 370, 664, 1272, 2350, 4466, 8372, 15926, 30105, 57390, 109202, 208738, 398985, 764906, 1467370, 2820770, 5427543, 10459456, 20176561, 38969684, 75339232, 145804978, 282429242, 547573768, 1062501151, 2063317650
Offset: 1

Views

Author

N. J. A. Sloane, Nov 02 2000

Keywords

References

  • P. J. Cameron, Some counting problems related to permutation groups, Discrete Math., 225 (2000), 77-92.

Links

Programs

  • Maple
    with(numtheory): ietr:= proc(p) local a, c; c:= proc(n) option remember; local j; n*p(n)-add(c(j)*p(n-j), j=1..n-1) end; a:=proc(n) option remember; local d; `if`(n=0,1, add(mobius(n/d)*c(d), d=divisors(n))/n) end end: a:= ietr(n-> add(phi(d) *2^(n/d)/2/n, d=select(m-> modp(m,2)=1, divisors(n)))): seq(a(n), n=1..40); # Alois P. Heinz, Sep 08 2008
# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(A000016):
seq(a(n), n = 1..39); # Peter Luschny, Nov 21 2022
  • Mathematica
    ietr[p_] := Module[{a, c}, c[n_] := c[n] = Module[{j}, n*p[n] - Sum[c[j]*p[n-j], {j, 1, n-1}]]; a[n_] := a[n] = Module[{d}, If[n == 0, 1, Sum[MoebiusMu[n/d]*c[d], {d, Divisors[n]}]/n]]; a]; a = ietr[Function[n, Sum[EulerPhi[d]*2^(n/d)/2/n, {d, Select[Divisors[n], OddQ]}]]]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jan 17 2014, after Alois P. Heinz *)

Extensions

Better definition and more terms from Vladeta Jovovic, Mar 13 2008

A131764 Inverse Euler transform of central binomial coefficients A000984.

Original entry on oeis.org

1, 2, 3, 10, 30, 102, 335, 1170, 4080, 14560, 52377, 190650, 698870, 2581110, 9586395, 35791358, 134215680, 505290270, 1908866960, 7233629130, 27487764474, 104715392730, 399822314775, 1529755308210, 5864061663920, 22517998136832, 86607683851185, 333599972392960, 1286742745883790, 4969489243995030, 19215358392200893, 74382032555280450, 288230376084602880
Offset: 0

Views

Author

F. Chapoton, Oct 04 2007

Keywords

Comments

This is the sequence of dimensions of a free Lie algebra on some specific set of generators.

Examples

			2*x + 3*x^2 + 10*x^3 + 30*x^4 + 102*x^5 + 335*x^6 + 1170*x^7 + 4080*x^8 + ...
(1-x)^(-2)*(1-x^2)^(-3)*(1-x^3)^(-10)*(1-x^4)^(-30)*(1-x^5)^(-102) = 1 + 2*x + 6*x^2 + 20*x^3 + 70*x^4 + 252*x^5 + ... .

Links

Crossrefs

Cf. A022553, A000984, A000108.

Programs

  • Maple
    # The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(n -> binomial(2*n, n)):
seq(a(n), n = 0..32); # Peter Luschny, Nov 21 2022
  • Mathematica
    a[n_] := (1/n)*DivisorSum[n, MoebiusMu[n/#]*2^(2*#-1)&]; Table[a[n], {n, 1, 32}] (* Jean-François Alcover, Feb 20 2017 *)
  • MuPAD
    a(n):=proc(n) begin 1/n*_plus(moebius(n/d)*2^(2*d-1)$d in divisors(n)) end;
  • PARI
    a(n)=sumdiv(n,d,1/n*moebius(n/d)*2^(d*2-1)); /* Joerg Arndt, Jul 06 2011 */
  • PARI
    {a(n) = local(A); if( n<1, 0, A = sqrt(1 - 4*x + x * O(x^n)); for( k=1, n-1, A *= (1 - x^k + x * O(x^n))^ polcoeff( A, k)); -polcoeff( A, n))} /* Michael Somos, Apr 01 2012 */

Formula

a(n) = (1/n) * Sum_{d|n} moebius(n/d)*2^(2*d-1) for n > 0, a(0) = 1.
a(n) ~ 2^(2*n-1) / n. - Vaclav Kotesovec, Oct 09 2019

Extensions

More explicit definition from Michael Somos, Apr 01 2012. - N. J. A. Sloane, Feb 20 2017

A358452 The inverse Euler transform of p(n) = n if n is prime, otherwise 1.

Original entry on oeis.org

1, 1, 1, 1, -3, 3, -3, 5, -8, 5, -11, 36, -45, 41, -72, 142, -223, 311, -493, 851, -1243, 1823, -3204, 5336, -7906, 12083, -20134, 33133, -51685, 81568, -133556, 215363, -340155, 547916, -895895, 1442323, -2300704, 3718260, -6056908, 9787064, -15755664, 25541623
Offset: 0

Views

Author

Peter Luschny, Nov 21 2022

Keywords

Comments

Conjecture: signum(a(n)) + (-1)^n = 0 for n >= 3.

Links

Crossrefs

Cf. A089026, A219224.

Programs

  • Maple
    # The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(n -> ifelse(isprime(n), n, 1)):
seq(a(n), n = 0..41);
# Using EULERi the sequence is returned without a(0) and has offset 1.
f := n -> ifelse(isprime(n), n, 1): EULERi([seq(f(n), n = 1..41)]);
Previous Showing 11-17 of 17 results.

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

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