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 1-5 of 5 results.

A318917 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, 8, 38, 262, 1732, 16144, 153596, 1660796, 19415384, 264084064, 3664187848, 57366995272, 936097392752, 16131362629568, 302946516251408, 6034409270818576, 125044362929875744, 2756094464546395264, 63280996793936902496
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 05 2018

Keywords

Links

Crossrefs

Cf. A088009, A318769, A318696.
Cf. A000262, A305127, A318695.
Cf. A053529, A028342.
Cf. A000010 (phi), A008683 (mu).

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[Sum[EulerPhi[k]*x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[EulerPhi[k]* a[n-k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 20}]
  • PARI
    a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, eulerphi(k)*a(n-k)/(n-k)!)); \\ Seiichi Manyama, Apr 29 2022

Formula

a(n)/n! ~ 3^(1/4) * exp(2*sqrt(6*n)/Pi) / (Pi * 2^(3/4) * n^(3/4)).
E.g.f.: Product_{k>=1} 1 / (1 - x^k)^f(k), where f(k) = (1/k) * Sum_{j=1..k} mu(gcd(k,j)). - Ilya Gutkovskiy, Aug 17 2021
a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} phi(k) * a(n-k)/(n-k)!. - Seiichi Manyama, Apr 29 2022
E.g.f.: exp( Sum_{n>=1} (mu(n)/n) * x^n/(1 - x^n) ), where mu(n) = A008683(n). - Paul D. Hanna, Jun 24 2023

A318695 Expansion of e.g.f. Product_{i>=1, j>=1} 1/(1 - x^(i*j))^(1/(i*j)).

Original entry on oeis.org

1, 1, 4, 16, 106, 658, 6088, 51952, 592828, 6577948, 88213744, 1173121024, 18663391096, 289030343704, 5157010548064, 92428084599232, 1848308567352592, 37038307949425168, 822602470902709312, 18285742807660340992, 444405771941314880416, 10883864256927386369056, 286778106663948874858624
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 31 2018

Keywords

Links

Crossrefs

Cf. A000005, A006171, A007425, A028342, A280540, A305127, A318696, A318977.

Programs

  • Maple
    seq(n!*coeff(series(mul(1/(1-x^k)^(tau(k)/k),k=1..100),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 09 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[Product[Product[1/(1 - x^(i j))^(1/(i j)), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[0, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[DivisorSigma[0, d], {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]

Formula

E.g.f.: Product_{k>=1} 1/(1 - x^k)^(tau(k)/k), where tau = number of divisors (A000005).
E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} tau(d) ) * x^k/k).

A318977 Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(tau(k)/k), where tau is A000005.

Original entry on oeis.org

1, 2, 8, 44, 292, 2296, 21472, 221168, 2554544, 32617952, 452957056, 6788855872, 110098330048, 1900192498304, 34971968379392, 683452436531456, 14097619892177152, 306168410773570048, 6998327049216231424, 167369475021548506112, 4187842602663179396096
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 06 2018

Keywords

Comments

Convolution of A318696 and A318695.

Links

Crossrefs

Cf. A318695, A318696, A318814, A318976.
Cf. A318975, A301555, A301554.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(DivisorSigma[0, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

A318769 Expansion of e.g.f. Product_{k>=1} (1 + x^k)^(sigma(k)/k), where sigma(k) is the sum of the divisors of k.

Original entry on oeis.org

1, 1, 3, 17, 83, 639, 5749, 53227, 561273, 7216577, 94292531, 1352253561, 21657812923, 359338829407, 6460367397093, 126124578755939, 2527688612931569, 54137820027005697, 1236730462664172643, 29137619131277727457, 725282418459957414051, 18981526480933601454911
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 03 2018

Keywords

Comments

a(n)/n! is the weigh transform of [1, 3/2, 4/3, 7/4, 6/5, ... = sums of reciprocals of divisors of 1, 2, 3, 4, 5, ...].

Links

Crossrefs

Cf. A000203, A017665, A017666, A168243, A192065, A288417, A305127, A318696, A318814.

Programs

  • Maple
    with(numtheory): a := proc(n) option remember; `if`(n = 0, 1, add(add(-(-1)^(j/d)*sigma(d), d = divisors(j))*a(n-j), j = 1..n)/n) end proc; seq(n!*a(n), n = 0..20); # Vaclav Kotesovec, Sep 04 2018
  • Mathematica
    nmax = 21; CoefficientList[Series[Product[(1 + x^k)^(DivisorSigma[1, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k + 1) x^(j k)/(j k (1 - x^(j k))), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d DivisorSigma[-1, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 21}]
nmax = 21; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[1, k]/k, j]*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; Take[CoefficientList[s, x], nmax + 1] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 03 2018 *)

Formula

E.g.f.: Product_{k>=1} (1 + x^k)^(A017665(k)/A017666(k)).
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(j*k*(1 - x^(j*k)))).
log(a(n)/n!) ~ sqrt(n/2) * Pi^2 / 3. - Vaclav Kotesovec, Sep 04 2018
a(n)/n! ~ c * exp(sqrt(n/2)*Pi^2/3) / n^(3/4 + log(2)/4), where c = 0.15653645678497413538057076667218805302154965061194080137... - Vaclav Kotesovec, Sep 05 2018

A318967 Expansion of e.g.f. Product_{i>=1, j>=1, k>=1} (1 + x^(i*j*k))^(1/(i*j*k)).

Original entry on oeis.org

1, 1, 3, 15, 69, 477, 4167, 34731, 333225, 4058073, 48535659, 638782119, 9690930477, 146665611765, 2428164153711, 44904494549763, 820664075440593, 16238018609968689, 350155700132388435, 7568774583230565567, 175171222712837235861, 4318996957424273510541, 107317465474650443023383
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 06 2018

Keywords

Links

Crossrefs

Cf. A000005, A007425, A168243, A280473, A318414, A318696, A318768, A318966.

Programs

  • Maple
    a:=series(mul(mul(mul((1+x^(i*j*k))^(1/(i*j*k)),k=1..55),j=1..55),i=1..55),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[Product[Product[Product[(1 + x^(i j k))^(1/(i j k)), {i, 1, nmax}], {j, 1, nmax}], {k, 1, nmax} ], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Product[(1 + x^k)^(Sum[DivisorSigma[0, d], {d, Divisors[k]}]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (n - 1)! Sum[Sum[(-1)^(k/d + 1) Sum[DivisorSigma[0, j], {j, Divisors[d]}], {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]

Formula

E.g.f.: Product_{k>=1} (1 + x^k)^(tau_3(k)/k), where tau_3 = A007425.
E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1) * Sum_{j|d} tau(j) ) * x^k/k), where tau = number of divisors (A000005).
Showing 1-5 of 5 results.

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