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 1-8 of 8 results.

A299069 Expansion of Product_{k>=1} (1 + x^k)^phi(k), where phi() is the Euler totient function (A000010).

Original entry on oeis.org

1, 1, 1, 3, 4, 8, 11, 19, 30, 44, 69, 103, 157, 229, 341, 491, 722, 1038, 1488, 2128, 3015, 4267, 5989, 8407, 11713, 16289, 22523, 31097, 42729, 58569, 80003, 108957, 147983, 200383, 270693, 364631, 490105, 656961, 878775, 1172653, 1561626, 2074982, 2751648, 3641536, 4810009, 6341365, 8344967
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 09 2018

Keywords

Links

Crossrefs

Cf. A000010, A061255, A107742, A159929, A192065, A318975.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(numtheory[phi](i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 09 2018
  • Mathematica
    nmax = 46; CoefficientList[Series[Product[(1 + x^k)^EulerPhi[k], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d EulerPhi[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 46}]

Formula

G.f.: Product_{k>=1} (1 + x^k)^A000010(k).
a(n) ~ exp(3^(5/3) * Zeta(3)^(1/3) * n^(2/3) / (2*Pi^(2/3))) * Zeta(3)^(1/6) / (2^(1/3) * 3^(1/6) * Pi^(5/6) * n^(2/3)). - Vaclav Kotesovec, Mar 23 2018

A300011 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k/k!), where phi() is the Euler totient function (A000010).

Original entry on oeis.org

1, 1, 2, 6, 20, 80, 362, 1820, 10084, 60522, 391864, 2714514, 20001700, 156107224, 1284705246, 11112088358, 100698613720, 953478331288, 9410963022318, 96614921664444, 1029705968813656, 11373102766644372, 129972789566984682, 1534638410054873892, 18696544357738885720
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 09 2018

Keywords

Comments

Exponential transform of A000010.

Examples

			E.g.f.: A(x) = 1 + x/1! + 2*x^2/2! + 6*x^3/3! + 20*x^4/4! + 80*x^5/5! + 362*x^6/6! + 1820*x^7/7! + ...

Links

Crossrefs

Cf. A000010, A050392, A159929, A274804, A295739.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      binomial(n-1, j-1)*numtheory[phi](j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 09 2018
  • Mathematica
    nmax = 24; CoefficientList[Series[Exp[Sum[EulerPhi[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[EulerPhi[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 24}]
  • PARI
    a(n) = if(n==0, 1, sum(k=1, n, eulerphi(k)*binomial(n-1, k-1)*a(n-k))); \\ Seiichi Manyama, Feb 27 2022

Formula

E.g.f.: exp(Sum_{k>=1} A000010(k)*x^k/k!).
a(0) = 1; a(n) = Sum_{k=1..n} phi(k) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Feb 27 2022

A340995 Triangle T(n,k) whose k-th column is the k-fold self-convolution of the Euler totient function phi; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 5, 3, 1, 0, 4, 8, 9, 4, 1, 0, 2, 16, 19, 14, 5, 1, 0, 6, 20, 42, 36, 20, 6, 1, 0, 4, 36, 72, 89, 60, 27, 7, 1, 0, 6, 44, 134, 184, 165, 92, 35, 8, 1, 0, 4, 68, 210, 376, 391, 279, 133, 44, 9, 1, 0, 10, 76, 348, 688, 880, 738, 441, 184, 54, 10, 1
Offset: 0

Views

Author

Alois P. Heinz, Feb 01 2021

Keywords

Examples

			Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  2,  2,   1;
  0,  2,  5,   3,   1;
  0,  4,  8,   9,   4,   1;
  0,  2, 16,  19,  14,   5,   1;
  0,  6, 20,  42,  36,  20,   6,   1;
  0,  4, 36,  72,  89,  60,  27,   7,   1;
  0,  6, 44, 134, 184, 165,  92,  35,   8,  1;
  0,  4, 68, 210, 376, 391, 279, 133,  44,  9,  1;
  0, 10, 76, 348, 688, 880, 738, 441, 184, 54, 10, 1;
  ...

Links

Crossrefs

Columns k=0-2 give (offsets may differ): A000007, A000010, A065093.
Row sums give A159929.
T(2n,n) gives A340994.

Programs

  • Maple
    T:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
      `if`(k=1, `if`(n=0, 0, numtheory[phi](n)), (q->
       add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);
  • Mathematica
    T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0],
     If[k == 1, If[n == 0, 0, EulerPhi[n]], With[{q = Quotient[k, 2]},
     Sum[T[j, q]*T[n - j, k - q], {j, 0, n}]]]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 13 2021, after Alois P. Heinz *)

Formula

T(n,k) = [x^n] (Sum_{j>=1} phi(j)*x^j)^k.

A319111 Expansion of Product_{k>=1} 1/(1 - phi(k)*x^k), where phi = Euler totient function (A000010).

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 22, 38, 63, 105, 174, 278, 447, 707, 1122, 1766, 2729, 4213, 6482, 9880, 15069, 22799, 34290, 51378, 76777, 114365, 169324, 250162, 368505, 540575, 792042, 1154798, 1680385, 2439101, 3530308, 5103380, 7349875, 10564955, 15155752, 21696072, 31007949, 44199845
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 10 2018

Keywords

Links

Crossrefs

Cf. A000010, A061255, A159929, A318917.
Cf. A279784, A306327, A316961.

Programs

  • Maple
    with(numtheory): a:=series(mul(1/(1-phi(k)*x^k),k=1..50),x=0,42): seq(coeff(a,x,n),n=0..41); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 41; CoefficientList[Series[Product[1/(1 - EulerPhi[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 41; CoefficientList[Series[Exp[Sum[Sum[EulerPhi[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d EulerPhi[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 41}]

Formula

G.f.: exp(Sum_{k>=1} Sum_{j>=1} phi(j)^k*x^(j*k)/k).
From Vaclav Kotesovec, Feb 08 2019: (Start)
a(n) ~ c * 2^(2*n/5), where
c = 18827.6460615531202942792897255332975807324818737172163... if mod(n,5) = 0
c = 18827.5079339024144115146595255453426552477117955925738... if mod(n,5) = 1
c = 18827.4967567108036710998657106724179082561779712900405... if mod(n,5) = 2
c = 18827.4818413568083742650057347700058389606441225811016... if mod(n,5) = 3
c = 18827.4547665561882994953942505862213438332903500716893... if mod(n,5) = 4
(End)

A352887 Expansion of e.g.f. 1/(1 - Sum_{k>=1} phi(k)*x^k/k!), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 1, 3, 14, 84, 634, 5740, 60626, 731852, 9938670, 149966116, 2489148386, 45070961740, 884107377360, 18676602726734, 422721143355808, 10205605681874952, 261789688633794528, 7110331886095458918, 203848868169846041430, 6151813078359073154568
Offset: 0

Views

Author

Seiichi Manyama, Apr 07 2022

Keywords

Crossrefs

Cf. A000010, A159929, A300011, A318811.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, N, eulerphi(k)*x^k/k!))))
  • PARI
    a(n) = if(n==0, 1, sum(k=1, n, eulerphi(k)*binomial(n, k)*a(n-k)));

Formula

a(0) = 1; a(n) = Sum_{k=1..n} phi(k) * binomial(n,k) * a(n-k).

A307073 Expansion of 1/(1 - Sum_{k>=1} mu(k)^2*x^k/(1 - x^k)^2).

Original entry on oeis.org

1, 1, 4, 11, 33, 94, 279, 803, 2348, 6823, 19879, 57834, 168405, 490125, 1426824, 4153197, 12089787, 35191868, 102440785, 298194567, 868017488, 2526715121, 7355031727, 21409798576, 62321907805, 181413177769, 528076639862, 1537181201003, 4474589318797, 13025106833162, 37914855831345
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 22 2019

Keywords

Comments

Invert transform of A001615.

Crossrefs

Cf. A001615, A008683, A156303, A159929, A301594.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[1/(1 - Sum[MoebiusMu[k]^2 x^k/(1 - x^k)^2, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 30}]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A001615(k)*a(n-k).

A307075 Expansion of 1/(1 - Sum_{k>=1} mu(k)*x^k*(1 + x^k)/(1 - x^k)^3).

Original entry on oeis.org

1, 1, 4, 15, 47, 160, 517, 1721, 5668, 18687, 61687, 203448, 671253, 2214377, 7305308, 24100319, 79506903, 262294336, 865310405, 2854666385, 9417565852, 31068622271, 102495625503, 338133855032, 1115506197957, 3680063534409, 12140557957708, 40051794232519, 132131177728807
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 22 2019

Keywords

Comments

Invert transform of A007434.

Crossrefs

Cf. A007434, A008683, A159929, A301875, A301876.

Programs

  • Mathematica
    nmax = 28; CoefficientList[Series[1/(1 - Sum[MoebiusMu[k] x^k (1 + x^k)/(1 - x^k)^3, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Sum[MoebiusMu[k/d] d^2, {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 28}]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A007434(k)*a(n-k).

A353192 Expansion of e.g.f. 1/(1 - Sum_{k>=1} phi(k) * x^k / k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 1, 3, 16, 110, 986, 10202, 126288, 1770120, 27939192, 489658632, 9455296896, 198951693360, 4537680805776, 111426422418768, 2931467216681856, 82273083792879744, 2453340521239749504, 77458777017799833216, 2581489882182061744128
Offset: 0

Views

Author

Seiichi Manyama, Apr 29 2022

Keywords

Crossrefs

Cf. A000010, A074930, A159929, A318917, A352887.

Programs

  • Mathematica
    phi[k_] := phi[k] = EulerPhi[k]; a[0] = 1; a[n_] := a[n] = Sum[(k - 1)! * phi[k] * Binomial[n, k] * a[n - k], {k, 1, n}]; Array[a, 20, 0] (* Amiram Eldar, Apr 30 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, N, eulerphi(k)*x^k/k))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (j-1)!*eulerphi(j)*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A074930(k) * binomial(n,k) * a(n-k).
Showing 1-8 of 8 results.

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