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 1-2 of 2 results.

A219265 O.g.f. satisfies: A(x) = Sum_{n>=0} A(n^2*x)^n * (n^2*x)^n/n! * exp(-n^2*x*A(n^2*x)).

Original entry on oeis.org

1, 1, 8, 160, 6918, 609469, 106947753, 37651271215, 26931993643529, 39243099256414069, 116654228928308598913, 710224935200206160129234, 8867331728829780268501045551, 227187317486051730833557991305666, 11969414396907448200529521385052444890
Offset: 0

Views

Author

Paul D. Hanna, Nov 16 2012

Keywords

Comments

Compare to the LambertW identity:
Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

Examples

			O.g.f.: A(x) = 1 + x + 8*x^2 + 160*x^3 + 6918*x^4 + 609469*x^5 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^4*x^2*A(2^2*x)^2/2!*exp(-2^2*x*A(2^2*x)) + 3^6*x^3*A(3^2*x)^3/3!*exp(-3^2*x*A(3^2*x)) + 4^8*x^4*A(4^2*x)^4/4!*exp(-4^2*x*A(4^2*x)) + 5^10*x^5*A(5^2*x)^5/5!*exp(-5^2*x*A(5^2*x)) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A219264, A218681, A218672, A219184, A217900.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^(2*k)*x^k*subst(A,x,k^2*x)^k/k!*exp(-k^2*x*subst(A,x,k^2*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A219343 O.g.f. satisfies: A(x) = Sum_{n>=0} A(n*x)^n * (n^3*x)^n/n! * exp(-n^3*x*A(n*x)).

Original entry on oeis.org

1, 1, 32, 3183, 650929, 226009218, 119298668857, 89086101638412, 89480710389500666, 116491795770107486363, 191172400354899371561288, 387419202671209086703674709, 956322827450633453264262285623, 2859815748552720894795327258080881, 10430012061189048036456303441601971435
Offset: 0

Views

Author

Paul D. Hanna, Nov 18 2012

Keywords

Comments

Compare to the LambertW identity:
Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

Examples

			O.g.f.: A(x) = 1 + x + 32*x^2 + 3183*x^3 + 650929*x^4 + 226009218*x^5 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^6*x^2*A(2*x)^2/2!*exp(-2^3*x*A(2*x)) + 3^9*x^3*A(3*x)^3/3!*exp(-3^3*x*A(3*x)) + 4^12*x^4*A(4*x)^4/4!*exp(-4^3*x*A(4*x)) + 5^15*x^5*A(5*x)^5/5!*exp(-5^3*x*A(5*x)) +...
simplifies to a power series in x with integer coefficients.

Crossrefs

Cf. A219264, A219265, A218681, A218672, A219184, A219342, A217900.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^(3*k)*x^k*subst(A,x,k*x)^k/k!*exp(-k^3*x*subst(A,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
Showing 1-2 of 2 results.

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