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.

A191803 G.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^(5*n^2).

Original entry on oeis.org

1, 1, 6, 61, 791, 11701, 188462, 3225915, 57840755, 1076423857, 20666351126, 407645638428, 8237858879315, 170229866493435, 3592746391559133, 77393340642273491, 1701286171473636404, 38169860244429063080
Offset: 0

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Author

Paul D. Hanna, Jun 16 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 61*x^3 + 791*x^4 + 11701*x^5 + 188462*x^6 +...
where the g.f. satisfies:
A(x) = 1 + x*A(x)^5 + x^2*A(x)^20 + x^3*A(x)^45 + x^4*A(x)^80 +...+ x^n*A(x)^(5*n^2) +...
		

Crossrefs

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*(A+x*O(x^n))^(5*m^2)));polcoeff(A,n)}

Formula

Let A = g.f. A(x), then A satisfies:
(1) A = Sum_{n>=0} x^n*A^(5*n)*Product_{k=1..n} (1-x*A^(20*k-15))/(1-x*A^(20*k-5));
(2) A = 1/(1- A^5*x/(1- A^5*(A^10-1)*x/(1- A^25*x/(1- A^15*(A^20-1)*x/(1- A^45*x/(1- A^25*(A^30-1)*x/(1- A^65*x/(1- A^35*(A^40-1)*x/(1- ...))))))))) (continued fraction);
due to a q-series identity and an identity of a partial elliptic theta function, respectively.

A191801 G.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^(3*n^2).

Original entry on oeis.org

1, 1, 4, 28, 251, 2573, 28813, 343833, 4308210, 56154805, 756731761, 10499096630, 149551069156, 2182935186698, 32613646656198, 498420592612153, 7790219357236805, 124545937719356873, 2037614647316548891, 34134979366157116560
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 28*x^3 + 251*x^4 + 2573*x^5 + 28813*x^6 +...
where the g.f. satisfies:
A(x) = 1 + x*A(x)^3 + x^2*A(x)^12 + x^3*A(x)^27 + x^4*A(x)^48 +...+ x^n*A(x)^(3*n^2) +...
		

Crossrefs

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*(A+x*O(x^n))^(3*m^2)));polcoeff(A,n)}

Formula

Let A = g.f. A(x), then A satisfies:
(1) A = Sum_{n>=0} x^n*A^(3*n)*Product_{k=1..n} (1-x*A^(12*k-9))/(1-x*A^(12*k-3));
(2) A = 1/(1- A^3*x/(1- A^3*(A^6-1)*x/(1- A^15*x/(1- A^9*(A^12-1)*x/(1- A^27*x/(1- A^15*(A^18-1)*x/(1- A^39*x/(1- A^21*(A^24-1)*x/(1- ...))))))))) (continued fraction);
due to a q-series identity and an identity of a partial elliptic theta function, respectively.

A191802 G.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^(4*n^2).

Original entry on oeis.org

1, 1, 5, 43, 473, 5942, 81393, 1186342, 18132473, 287948903, 4722077279, 79636530163, 1377304530677, 24382127678100, 441294262119031, 8160739579770316, 154169018332135841, 2975846752734820345, 58718914018159811186
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 43*x^3 + 473*x^4 + 5942*x^5 + 81393*x^6 +...
where the g.f. satisfies:
A(x) = 1 + x*A(x)^4 + x^2*A(x)^16 + x^3*A(x)^36 + x^4*A(x)^64 +...+ x^n*A(x)^(4*n^2) +...
		

Crossrefs

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*(A+x*O(x^n))^(4*m^2)));polcoeff(A,n)}

Formula

Let A = g.f. A(x), then A satisfies:
(1) A = Sum_{n>=0} x^n*A^(4*n)*Product_{k=1..n} (1-x*A^(16*k-12))/(1-x*A^(16*k-4));
(2) A = 1/(1- A^4*x/(1- A^4*(A^8-1)*x/(1- A^20*x/(1- A^12*(A^16-1)*x/(1- A^36*x/(1- A^20*(A^24-1)*x/(1- A^52*x/(1- A^28*(A^32-1)*x/(1- ...))))))))) (continued fraction);
due to a q-series identity and an identity of a partial elliptic theta function, respectively.

A191804 G.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^(6*n^2).

Original entry on oeis.org

1, 1, 7, 82, 1221, 20718, 382315, 7489683, 153551487, 3264643144, 71545452946, 1609541143713, 37065029428453, 872037022019930, 20935244357544798, 512498682139660135, 12790021472251565047, 325439165493879484025
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 7*x^2 + 82*x^3 + 1221*x^4 + 20718*x^5 + 382315*x^6 +...
where the g.f. satisfies:
A(x) = 1 + x*A(x)^6 + x^2*A(x)^24 + x^3*A(x)^54 + x^4*A(x)^96 +...+ x^n*A(x)^(6*n^2) +...
		

Crossrefs

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*(A+x*O(x^n))^(6*m^2)));polcoeff(A,n)}

Formula

Let A = g.f. A(x), then A satisfies:
(1) A = Sum_{n>=0} x^n*A^(6*n)*Product_{k=1..n} (1-x*A^(24*k-18))/(1-x*A^(24*k-6));
(2) A = 1/(1- A^6*x/(1- A^6*(A^12-1)*x/(1- A^30*x/(1- A^18*(A^24-1)*x/(1- A^54*x/(1- A^30*(A^36-1)*x/(1- A^78*x/(1- A^42*(A^48-1)*x/(1- ...))))))))) (continued fraction);
due to a q-series identity and an identity of a partial elliptic theta function, respectively.
Showing 1-4 of 4 results.