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.

A156302 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)^2*x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 1, 5, 10, 30, 57, 152, 289, 676, 1304, 2809, 5335, 10961, 20487, 40329, 74476, 141914, 258094, 479638, 860025, 1563716, 2767982, 4940567, 8636563, 15173805, 26217392, 45416811, 77629455, 132800937, 224695510, 380079521, 637006921
Offset: 0

Views

Author

Paul D. Hanna, Feb 08 2009

Keywords

Comments

Compare with g.f. for partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 10*x^3 + 30*x^4 + 57*x^5 + 152*x^6 +...
log(A(x)) = x + 3^2*x^2/2 + 4^2*x^3/3 + 7^2*x^4/4 + 6^2*x^5/5 + 12^2*x^6/6 +...
Also log(A(x)) = (x + 3*x^2 + 4*x^3 + 7*x^4 +...+ sigma(k)*x^k +...)/1 +
(3*x^2 + 7*x^4 + 12*x^6 + 15*x^8 + 18*x^10 +...+ sigma(2*k)*x^(2*k) +...)/2 +
(4*x^3 + 12*x^6 + 13*x^9 + 28*x^12 + 24*x^15 +...+ sigma(3*k)*x^(3*k) +...)/3 +
(7*x^4 + 15*x^8 + 28*x^12 + 31*x^16 + 42*x^20 +...+ sigma(4*k)*x^(4*k) +...)/4 +
(6*x^5 + 18*x^10 + 24*x^15 + 42*x^20 + 31*x^25 +...+ sigma(5*k)*x^(5*k) +...)/5 +...

Links

Crossrefs

Cf. A000203 (sigma), A000041 (partitions), A072861, A178933, A205797, A382125.

Programs

  • Mathematica
    nmax = 40; $RecursionLimit -> Infinity; a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSigma[1, k]^2*a[n-k], {k, 1, n}]/n]; Table[a[n], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 29 2024 *)
  • PARI
    {a(n)=polcoeff(exp(sum(k=1,n,sigma(k)^2*x^k/k)+x*O(x^n)),n)}
  • PARI
    {a(n)=if(n==0,1,(1/n)*sum(k=1,n,sigma(k)^2*a(n-k)))}
  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n+1,sum(k=1,n\m,sigma(m*k)*x^(m*k)/m)+x*O(x^n))),n)}

Formula

a(n) = (1/n)*Sum_{k=1..n} sigma(k)^2*a(n-k) for n>0, with a(0) = 1.
Euler transform of A060648. [From Vladeta Jovovic, Feb 14 2009]
It appears that G.f.: A(x)=prod(n=1,infinity, E(x^n)^(-A001615(n))) where E(x) = prod(n=1,infinity,1-x^n). [From Joerg Arndt, Dec 30 2010]
G.f.: exp( Sum_{n>=1} Sum_{k>=1} sigma(n*k) * x^(n*k) / n ). [From Paul D. Hanna, Jan 23 2012]
log(a(n)) ~ 3*(5*zeta(3))^(1/3) * n^(2/3) / 2. - Vaclav Kotesovec, Oct 29 2024

A178933 Generating function exp( sum(n>=1, sigma(n)^3*x^n/n ) ).

Original entry on oeis.org

1, 1, 14, 35, 205, 521, 2507, 6709, 26712, 73834, 262431, 724537, 2384988, 6552033, 20289864, 55244988, 163342701, 439201501, 1251532060, 3321188863, 9177476977, 24028568664, 64709650590, 167153761523, 440300702427, 1122562426240, 2900254892900, 7301575351055, 18544013542057
Offset: 0

Views

Author

Joerg Arndt, Dec 30 2010

Keywords

Comments

Compare with g.f. for partition numbers A000041: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.
Similarly, exp( Sum_{n>=1} sigma(n)^2*x^n/n ) gives A156302.

Examples

			G.f.: A(x) = 1 + x + 14*x^2 + 35*x^3 + 205*x^4 + 521*x^5 + 2507*x^6 +...
such that, by definition,
log(A(x)) = x + 3^3*x^2/2 + 4^3*x^3/3 + 7^3*x^4/4 + 6^3*x^5/5 + 12^3*x^6/6 +...

Links

Crossrefs

Cf. A000203 (sigma), A000041 (partitions), A156302, A205797, A361147.

Programs

  • Mathematica
    nmax = 30; $RecursionLimit -> Infinity; a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSigma[1,k]^3 * a[n-k], {k, 1, n}]/n]; Table[a[n], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 30 2024 *)
  • PARI
    N=100;v=Vec(exp(sum(k=1,N,sigma(k)^3*x^k/k)+x*O(x^N)))
  • PARI
    a(n)=if(n==0, 1, (1/n)*sum(k=1, n, sigma(k)^3*a(n-k)))
  • PARI
    {a(n)=polcoeff(exp(sum(k=1, n, sigma(k)^3*x^k/k)+x*O(x^n)), n)} /* Paul D. Hanna */
  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n+1, sum(k=1, n\m, sigma(m*k)^2*x^(m*k)/m)+x*O(x^n))), n)} /* Paul D. Hanna */

Formula

a(0)=0 and a(n)=1/n*sum(k=1,n,sigma(k)^3*a(n-k)) for n>0.
G.f.: exp( Sum_{n>=1} Sum_{k>=1} sigma(n*k)^2 * x^(n*k) / n ). [Paul D. Hanna, Jan 31 2012]
From Vaclav Kotesovec, Oct 30 2024: (Start)
log(a(n)) ~ 2^(7/4) * c^(1/4) * Pi^(3/2) * zeta(3)^(1/4) * n^(3/4) / (3^(3/2) * 5^(1/4)), where c = Product_{primes p} (1 + 2/p^2 + 2/p^3 + 1/p^5) = 2.8359835743341928677044245715803848964089818378791769798895797934086403174189...
Equivalently, log(a(n)) ~ 3.2753680082113515869730831738879060384726246... * n^(3/4). (End)

A377509 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)^4 * x^k/k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 1, 2, 36, 234, 7290, 54540, 1408680, 23119740, 341788860, 11790437400, 231972879600, 8206299070200, 191673262380600, 6154270418696400, 206515993375692000, 6574758436640394000, 269828090984990538000, 9531096165082736244000, 411037724983993923816000
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 30 2024

Keywords

Links

Crossrefs

Cf. A318917, A377507, A377508.
Cf. A205797, A361148.

Programs

  • Mathematica
    nmax = 25; $RecursionLimit->Infinity; a[n_]:=a[n]=If[n==0, 1, Sum[EulerPhi[k]^4 * a[n-k], {k, 1, n}]/n];Table[a[n]*n!, {n, 0, nmax}]
nmax = 25; CoefficientList[Series[Exp[Sum[EulerPhi[k]^4 * x^k / k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!

Formula

log(a(n)/n!) ~ 5 * 3^(1/5) * c^(1/5) * n^(4/5) / 2^(7/5), where c = Product_{primes p} (1 - 4/p^2 + 6/p^3 - 4/p^4 + 1/p^5) = 0.286256471511560891173288340086638647956...
Showing 1-3 of 3 results.

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