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.

A219331 L.g.f.: -log(1 - Sum_{n>=1} x^(n^2)) = Sum_{n>=1} a(n)*x^n/n.

Original entry on oeis.org

1, 1, 1, 5, 6, 7, 8, 13, 28, 36, 45, 59, 92, 134, 186, 269, 375, 538, 761, 1080, 1520, 2157, 3060, 4339, 6181, 8750, 12394, 17554, 24912, 35322, 50066, 70957, 100596, 142665, 202278, 286790, 406520, 576347, 817142, 1158528, 1642461, 2328536, 3301283, 4680417, 6635688
Offset: 1

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Author

Paul D. Hanna, Apr 12 2013

Keywords

Comments

Limit a(n)/a(n+1) = 0.705346681379806989636379706393941505260078161512292870... is a real root of 1 = Sum_{n>=1} x^(n^2).

Examples

			L.g.f.: L(x) = x + x^2/2 + x^3/3 + 5*x^4/4 + 6*x^5/5 + 7*x^6/6 + 8*x^7/7 + 13*x^8/8 + 28*x^9/9 + 36*x^10/10 +...
where
exp(L(x)) = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 7*x^8 + 11*x^9 + 16*x^10 + 22*x^11 + 30*x^12 +...+ A006456(n)*x^n +...
exp(-L(x)) = 1 - x - x^4 - x^9 - x^16 - x^25 - x^36 +...+ -x^(n^2) +...

Links

Crossrefs

Cf. A224607, A224608, A006456.

Programs

  • PARI
    {a(n)=n*polcoeff(-log(1-sum(r=1,sqrtint(n+1),x^(r^2)+x*O(x^n))),n)}
for(n=1,50,print1(a(n),", "))

Formula

Logarithmic derivative of A006456, where A006456(n) is the number of compositions of n into sums of squares.

A224607 a(n) = A219331(n^2).

Original entry on oeis.org

1, 5, 28, 269, 6181, 286790, 26808447, 5037694829, 1902773895751, 1444565587750055, 2204357811343981558, 6761166975496300074014, 41682712965722542326438411, 516517498759950258411494666787, 12864972023450485679400300069493738
Offset: 1

Views

Author

Paul D. Hanna, Apr 12 2013

Keywords

Comments

A219331 is the logarithmic derivative of A006456, where A006456(n) is the number of compositions of n into sums of squares.

Examples

			L.g.f.: L(x) = x + 5*x^2/2 + 28*x^3/3 + 269*x^4/4 + 6181*x^5/5 + 286790*x^6/6 +...
where
exp(L(x)) = 1 + x + 3*x^2 + 12*x^3 + 81*x^4 + 1335*x^5 + 49309*x^6 + 3882180*x^7 +...+ A224608(n)*x^n +...

Crossrefs

Cf. A224608, A219331, A006456, A224366.

Programs

  • PARI
    {a(n)=n^2*polcoeff(-log(1-sum(r=1,n,x^(r^2)+x*O(x^(n^2)))),n^2)}
for(n=1,30,print1(a(n),", "))

Formula

Logarithmic derivative of A224608.

A224681 G.f.: exp( Sum_{n>=1} A224678(n^2) * x^n/n ).

Original entry on oeis.org

1, 1, 3, 19, 300, 11768, 1193594, 302611474, 188884066846, 288112683033594, 1069431906358800731, 9633610233639395592895, 210208585613243673600527636, 11095213297186302234251136888284, 1415095855034367649056280021793496073, 435753686684779400844511781608578944222819
Offset: 0

Views

Author

Paul D. Hanna, Apr 14 2013

Keywords

Comments

A224678 is the logarithmic derivative of A023361, where A023361(n) = number of compositions of n into positive triangular numbers.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 19*x^3 + 300*x^4 + 11768*x^5 + 1193594*x^6 +...
where
log(A(x)) = x + 5*x^2/2 + 49*x^3/3 + 1117*x^4/4 + 57181*x^5/5 + 7086833*x^6/6 +...+ A224678(n^2)*x^n/n +...

Crossrefs

Cf. A224678, A224680, A224608.

Programs

  • PARI
    {A224678(n)=n*polcoeff(-log(1-sum(r=1, sqrtint(2*n+1), x^(r*(r+1)/2)+x*O(x^n))), n)}
{a(n)=polcoeff(exp(sum(m=1, n, A224678(m^2)*x^m/m)+x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", "))

Formula

Logarithmic derivative yields A224680.
Showing 1-3 of 3 results.

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