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 1-4 of 4 results.

A205478 G.f.: exp( Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + d*x^n) ).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 8, 8, 14, 15, 24, 25, 43, 45, 69, 74, 113, 120, 187, 198, 291, 314, 452, 483, 720, 770, 1089, 1182, 1657, 1784, 2530, 2724, 3764, 4102, 5593, 6053, 8361, 9049, 12183, 13304, 17831, 19378, 26097, 28355, 37548, 41107, 54031, 58894, 78008, 85052
Offset: 0

Views

Author

Paul D. Hanna, Jan 27 2012

Keywords

Comments

Note: exp( Sum_{n>=1} (x^n/n) * Product_{d|n} (1 + x^n) ) does not yield an integer series.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 8*x^6 + 8*x^7 + ...
By definition:
log(A(x)) = x*(1+x) + x^2*(1+x^2)*(1+2*x^2)/2 + x^3*(1+x^3)*(1+3*x^3)/3 + x^4*(1+x^4)*(1+2*x^4)*(1+4*x^4)/4 + x^5*(1+x^5)*(1+5*x^5)/5 + x^6*(1+x^6)*(1+2*x^6)*(1+3*x^6)*(1+6*x^6)/6 + ...
Explicitly,
log(A(x)) = x + 3*x^2/2 + x^3/3 + 7*x^4/4 + x^5/5 + 15*x^6/6 + x^7/7 + 15*x^8/8 + 10*x^9/9 + 13*x^10/10 + ... + A205479(n)*x^n/n + ...

Crossrefs

Cf. A205479, A198296, A205476, A205480, A205482, A205484, A205486, A205488, A205490.

Programs

  • Mathematica
    max = 50; s = Exp[Sum[(x^n/n)*Product[1+d*x^n, {d, Divisors[n]}], {n, 1, max}]] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 23 2015 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, log(1+d*x^m+x*O(x^n)))))), n)}

Formula

Logarithmic derivative yields A205479.

A198299 L.g.f.: Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^n).

Original entry on oeis.org

1, 3, 4, 11, 6, 36, 8, 83, 49, 178, 12, 680, 14, 920, 714, 2707, 18, 7119, 20, 14166, 7844, 22564, 24, 94616, 3931, 106538, 88987, 306604, 30, 832606, 32, 1401715, 974736, 2228278, 150758, 9643703, 38, 9961532, 10363682, 28802278, 42, 78793604, 44, 123016344
Offset: 1

Views

Author

Paul D. Hanna, Jan 26 2012

Keywords

Comments

Forms the logarithmic derivative of A198296.

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 4*x^3/3 + 11*x^4/4 + 6*x^5/5 + 36*x^6/6 +...
such that, by definition:
L(x) = x/(1-x) + (x^2/2)/((1-x^2)*(1-2*x^2)) + (x^3/3)/((1-x^3)*(1-3*x^3)) + (x^4/4)/((1-x^4)*(1-2*x^4)*(1-4*x^4)) + (x^5/5)/((1-x^5)*(1-5*x^5)) + (x^6/6)/((1-x^6)*(1-2*x^6)*(1-3*x^6)*(1-6*x^6)) +...+ (x^n/n)/Product_{d|n} (1-d*x^n) +...
Also, we have the identity:
L(x) = (1 + x + x^2 + x^3 + x^4 + x^5 +...)*x
+ (1 + 3*x^2 + 7*x^4 + 15*x^6 + 31*x^8 +...)*x^2/2
+ (1 + 4*x^3 + 13*x^6 + 40*x^9 + 121*x^12 +...)*x^3/3
+ (1 + 7*x^4 + 35*x^8 + 155*x^12 + 651*x^16 +...)*x^4/4
+ (1 + 6*x^5 + 31*x^10 + 156*x^15 + 781*x^20 +...)*x^5/5
+ (1 + 12*x^6 + 97*x^12 + 672*x^18 + 4333*x^24 +...)*x^6/6 +...
+ exp( Sum_{k>=1} sigma(n,k)*x^(n*k)/k )*x^n/n +...
Exponentiation yields the g.f. of A198296:
exp(L(x)) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 8*x^5 + 17*x^6 + 22*x^7 +...

Links

Crossrefs

Cf. A198296 (exp), A198305.

Programs

  • PARI
    {a(n)=n*polcoeff(sum(m=1, n+1, x^m/m*exp(sum(k=1, n\m, sigma(m, k)*x^(m*k)/k)+x*O(x^n))), n)}
  • PARI
    {a(n)=n*polcoeff(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d,-log(1-d*x^m+x*O(x^n))))), n)}

Formula

L.g.f.: Sum_{n>=1} x^n/n * exp( Sum_{k>=1} sigma(n,k) * x^(n*k)/k ), where sigma(n,k) is the sum of the k-th powers of the divisors of n.

A198304 G.f.: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - n*x^d/d) ).

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 54, 148, 442, 1433, 5061, 19394, 80308, 357241, 1697870, 8577240, 45845235, 258198133, 1526631800, 9445795717, 60988643813, 409933740177, 2862338202947, 20723903238290, 155329193200741, 1203428108558453, 9624564394649845, 79357873429159078
Offset: 0

Views

Author

Paul D. Hanna, Jan 27 2012

Keywords

Comments

Logarithmic derivative yields A198305.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 21*x^5 + 54*x^6 + 148*x^7 +...
such that, by definition:
log(A(x)) = x/(1-x) + (x^2/2)/((1-2*x)*(1-x^2)) + (x^3/3)/((1-3*x)*(1-x^3)) + (x^4/4)/((1-4*x)*(1-2*x^2)*(1-x^4)) + (x^5/5)/((1-5*x)*(1-x^5)) + (x^6/6)/((1-6*x)*(1-3*x^2)*(1-2*x^3)*(1-x^6)) +...+ (x^n/n)/Product_{d|n} (1-n*x^d/d) +...
Explicitly, the logarithm begins:
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 51*x^5/5 + 159*x^6/6 + 519*x^7/7 + 1867*x^8/8 + 7234*x^9/9 +...+ A198305(n)*x^n/n +...

Crossrefs

Cf. A198305 (log), A198296.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*exp(sumdiv(m, d, -log(1-m*x^d/d+x*O(x^n)))))), n)}

A198301 G.f.: exp( Sum_{n>=1} (x^n/n) * Sum_{d|n} d*sigma(n/d,d) ).

Original entry on oeis.org

1, 1, 3, 5, 12, 18, 42, 62, 131, 206, 398, 610, 1203, 1810, 3358, 5260, 9471, 14518, 26182, 39906, 70320, 108849, 187251, 287525, 497288, 758860, 1286936, 1986352, 3330677, 5102712, 8560107, 13070327, 21685731, 33328561, 54744685, 83792111, 137817745, 210223967
Offset: 0

Views

Author

Paul D. Hanna, Jan 27 2012

Keywords

Comments

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

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 5*x^3 + 12*x^4 + 18*x^5 + 42*x^6 + 62*x^7 +...
where the logarithm begins:
log(A(x)) = x + 5*x^2/2 + 7*x^3/3 + 21*x^4/4 + 11*x^5/5 + 65*x^6/6 + 15*x^7/7 + 133*x^8/8 + 106*x^9/9 +...+ A198302(n)*x^n/n +...

Crossrefs

Cf. A198302 (log), A198296.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n+1,sumdiv(m, d, d*sigma(m/d,d))*x^m/m)+x*O(x^n)),n)}
  • 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

G.f.: exp( Sum_{n>=1} Sum_{k>=1} sigma(n,k) * x^(n*k)/n ).
Logarithmic derivative yields A198302.
Showing 1-4 of 4 results.

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