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.

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A206765 G.f.: Product_{n>=1} [ (1 - 3^n*x^n) / (1 - (n+3)^n*x^n) ]^(1/n).

Original entry on oeis.org

1, 1, 12, 87, 907, 8393, 118932, 1683990, 31209334, 635005549, 15054451057, 393600573337, 11466736952722, 363842430190308, 12564913404375244, 467483278911401155, 18670853023655302285, 795978439482823960066, 36093307429580735618893
Offset: 0

Views

Author

Paul D. Hanna, Feb 12 2012

Keywords

Comments

Here sigma(n,k) equals the sum of the k-th powers of the divisors of n.

Examples

			G.f.: A(x) = 1 + x + 12*x^2 + 87*x^3 + 907*x^4 + 8393*x^5 + 118932*x^6 +...
where the g.f. equals the product:
A(x) = (1-3*x)/(1-4*x) * ((1-3^2*x^2)/(1-5^2*x^2))^(1/2) * ((1-3^3*x^3)/(1-6^3*x^3))^(1/3) * ((1-3^4*x^4)/(1-7^4*x^4))^(1/4) * ((1-3^5*x^5)/(1-8^5*x^5))^(1/5) *...
The logarithm equals the l.g.f. of A206766:
log(A(x)) = x + 23*x^2/2 + 226*x^3/3 + 3039*x^4/4 + 33306*x^5/5 +...

Links

Crossrefs

Cf. A206766 (log), A205814, A205811, A206763.

Programs

  • Mathematica
    max = 19; p = Product[((1-3^n*x^n) / (1-(n+3)^n*x^n))^(1/n), {n, 1, max}] + O[x]^max; CoefficientList[p, x] (* Jean-François Alcover, Oct 08 2016 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*sum(k=1, m, binomial(m, k)*sigma(m, k)*3^(m-k))+x*O(x^n))), n)}
  • PARI
    {a(n)=polcoeff(prod(k=1, n, ((1-3^k*x^k)/(1-(k+3)^k*x^k +x*O(x^n)))^(1/k)), n)}
for(n=0,31,print1(a(n),", "))

Formula

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=1..n} binomial(n,k) * sigma(n,k) * 3^(n-k) ).
Logarithmic derivative yields A206766.
a(n) ~ exp(3) * n^(n-1). - Vaclav Kotesovec, Oct 08 2016

A206764 a(n) = Sum_{k=1..n} binomial(n,k) * sigma(n,k) * (-1)^(n-k).

Original entry on oeis.org

1, -1, 10, 79, 1026, 15686, 279938, 5771359, 134218243, 3487832974, 100000000002, 3138673052878, 106993205379074, 3937454749863382, 155568096631586820, 6568441588686506943, 295147905179352825858, 14063102470280932000757, 708235345355337676357634
Offset: 1

Views

Author

Paul D. Hanna, Feb 12 2012

Keywords

Comments

Here sigma(n,k) equals the sum of the k-th powers of the divisors of n.

Examples

			L.g.f.: L(x) = x - x^2/2 + 10*x^3/3 + 79*x^4/4 + 1026*x^5/5 + 15686*x^6/6 +...
Exponentiation yields the g.f. of A206763:
exp(L(x)) = 1 + x + 3*x^3 + 23*x^4 + 225*x^5 + 2824*x^6 + 42670*x^7 +...
Illustration of terms.
a(2) = -2*sigma(2,1) + 1*sigma(2,2) = -2*3 + 1*5 = -1;
a(3) = 3*sigma(3,1) - 3*sigma(3,2) + 1*sigma(3,3) = 3*4 - 3*10 + 1*28 = 10;
a(4) = -4*sigma(4,1) + 6*sigma(4,2) - 4*sigma(4,3) + 1*sigma(4,4) = -4*7 + 6*21 - 4*73 + 1*273 = 79.

Crossrefs

Cf. A206763 (exp), A205815, A205812.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k] * DivisorSigma[k, n] * (-1)^(n-k), {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Oct 25 2024 *)
  • PARI
    {a(n)=sum(k=1, n, binomial(n, k)*sigma(n, k)*(-1)^(n-k))}
  • PARI
    {a(n)=n*polcoeff(sum(k=1, n, (1/k)*log((1-(-x)^k)/(1-(k-1)^k*x^k +x*O(x^n)))), n)}
for(n=1,21,print1(a(n),", "))

Formula

a(n) ~ exp(-1) * n^n. - Vaclav Kotesovec, Oct 25 2024
Showing 1-2 of 2 results.

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