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.

A219261 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n*A(x^n)/n * Product_{k>=1} (1 + x^(n*k)*A(x^n)^k) ).

Original entry on oeis.org

1, 1, 3, 9, 33, 124, 503, 2089, 8960, 39142, 173978, 783347, 3567123, 16395199, 75966835, 354447193, 1663921966, 7853325055, 37244059607, 177388171005, 848148206917, 4069483589180, 19588001935380, 94559416543623, 457697632011720, 2220852281129195, 10800560004895426
Offset: 0

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Author

Paul D. Hanna, Nov 16 2012

Keywords

Comments

Compare to the dual g.f. G(x) of A219260:
G(x) = exp( Sum_{n>=1} x^n*G(x)^n/n * Product_{k>=1} (1 + x^(n*k)*G(x^k)^n) ).

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 33*x^4 + 124*x^5 + 503*x^6 + 2089*x^7 +...
where
log(A(x)) = x*A(x)/1*((1+x*A(x))*(1+x^2*A(x)^2)*(1+x^3*A(x)^3)*...) +
x^2*A(x^2)/2*((1+x^2*A(x^2))*(1+x^4*A(x^2)^2)*(1+x^6*A(x^2)^3)*...) +
x^3*A(x^3)/3*((1+x^3*A(x^3))*(1+x^6*A(x^3)^2)*(1+x^9*A(x^3)^3)*...) +
x^4*A(x^4)/4*((1+x^4*A(x^4))*(1+x^8*A(x^4)^2)*(1+x^12*A(x^4)^3)*...) +...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 19*x^3/3 + 89*x^4/4 + 396*x^5/5 + 1895*x^6/6 + 8989*x^7/7 + 43545*x^8/8 + 211645*x^9/9 + 1036560*x^10/10 +...

Crossrefs

Cf. A219232, A219263, A219260, A218153.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*subst(A, x, x^m +x*O(x^n))/m*prod(k=1, n\m+1, 1+x^(m*k)*subst(A^k, x, x^m +x*O(x^n)))))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

A219262 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n*A(x)^n/n / Product_{k>=1} (1 - x^(n*k)*A(x^k)^n) ).

Original entry on oeis.org

1, 1, 3, 11, 43, 179, 772, 3446, 15737, 73315, 346869, 1662593, 8055433, 39390631, 194145569, 963490818, 4810435187, 24145458246, 121771909878, 616744946740, 3135657413451, 15997777629900, 81877262767137, 420263466778106, 2162874177974610, 11158378099471260, 57696623896231494
Offset: 0

Views

Author

Paul D. Hanna, Nov 16 2012

Keywords

Comments

Compare to the dual g.f. G(x) of A219263:
G(x) = exp( Sum_{n>=1} x^n*G(x^n)/n / Product_{k>=1} (1 - x^(n*k)*G(x^n)^k) ).

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 43*x^4 + 179*x^5 + 772*x^6 + 3446*x^7 +...
where
log(A(x)) = x*A(x)/1/(1-x*A(x))*(1-x^2*A(x^2))*(1-x^3*A(x^3))*...) +
x^2*A(x)^2/2/((1-x^2*A(x)^2)*(1-x^4*A(x^2)^2)*(1-x^6*A(x^3)^2)*...) +
x^3*A(x)^3/3/((1-x^3*A(x)^3)*(1-x^6*A(x^2)^3)*(1-x^9*A(x^3)^3)*...) +
x^4*A(x)^4/4/((1-x^4*A(x)^4)*(1-x^8*A(x^2)^4)*(1-x^12*A(x^3)^4)*...) +...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 25*x^3/3 + 121*x^4/4 + 601*x^5/5 + 2999*x^6/6 + 15247*x^7/7 + 78057*x^8/8 + 403279*x^9/9 + 2095425*x^10/10 +...

Crossrefs

Cf. A218551, A219260, A219263.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*A^m/m/prod(k=1, n\m+1, 1-x^(m*k)*subst(A^m, x, x^k +x*O(x^n)))))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
Showing 1-2 of 2 results.

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