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

Views

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.

A224608 G.f.: exp( Sum_{n>=1} A219331(n^2)*x^n/n ).

Original entry on oeis.org

1, 1, 3, 12, 81, 1335, 49309, 3882180, 633703214, 212061201327, 144669917959584, 200541263416077021, 563631413420071614333, 3206926569346230863485855, 36897315109526505791310840932, 857701705296285206387609947414980, 40254707002970300021370965171570478599
Offset: 0

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

			G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 81*x^4 + 1335*x^5 + 49309*x^6 +...
where
log(A(x)) = x + 5*x^2/2 + 28*x^3/3 + 269*x^4/4 + 6181*x^5/5 + 286790*x^6/6 +...+ A219331(n^2)*x^n/n +...

Crossrefs

Cf. A224607, A219331, A006456, A224366.

Programs

  • PARI
    {A219331(n)=n*polcoeff(-log(1-sum(r=1,sqrtint(n+1),x^(r^2)+x*O(x^n))),n)}
{a(n)=polcoeff(exp(sum(m=1,n,A219331(m^2)*x^m/m)+x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))

Formula

Logarithmic derivative yields A224607, where A224607(n) = A219331(n^2).

A224680 a(n) = A224678(n^2).

Original entry on oeis.org

1, 5, 49, 1117, 57181, 7086833, 2109733585, 1508630963069, 2591308566579217, 10691434112980070315, 105957942450483004330197, 2522387398320711543274084153, 144235039901139444727535460625985, 19811186631607253937472121882634566325
Offset: 1

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

			L.g.f.: L(x) = x + 5*x^2/2 + 49*x^3/3 + 1117*x^4/4 + 57181*x^5/5 + 7086833*x^6/6 +...
where
exp(L(x)) = 1 + x + 3*x^2 + 19*x^3 + 300*x^4 + 11768*x^5 + 1193594*x^6 +...+ A224681(n)*x^n +...

Crossrefs

Cf. A224681, A224678, A224607.

Programs

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

Formula

Logarithmic derivative of A224681.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.