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.

A196150 G.f. satisfies A(x) = 1/Product_{n>=1} (1 - x^n*A(x)^2).

Original entry on oeis.org

1, 1, 4, 18, 92, 505, 2922, 17541, 108270, 682823, 4380942, 28504466, 187636994, 1247375147, 8362420498, 56471709841, 383790966537, 2622982116829, 18016055333571, 124296340608870, 860986586024343, 5985590694574930, 41749023026002831
Offset: 0

Views

Author

Paul D. Hanna, Sep 28 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 18*x^3 + 92*x^4 + 505*x^5 + 2922*x^6 + ...
where
(0) A(x) = 1/((1-x*A(x)^2) * (1-x^2*A(x)^2) * (1-x^3*A(x)^2) * ...).
(1) A(x) = 1 + x*A(x)^2/(1-x) + x^2*A(x)^4/((1-x)*(1-x^2)) + x^3*A(x)^6/((1-x)*(1-x^2)*(1-x^3)) + ...
(2) A(x) = 1 + x*A(x)^2/[(1-x)*(1-x*A(x)^2)] + x^4*A(x)^4/[(1-x)*(1-x^2)*(1-x*A(x)^2)*(1-x^2*A(x)^2)] + x^9*A(x)^6/[(1-x)*(1-x^2)*(1-x^3)*(1-x*A(x)^2)*(1-x^2*A(x)^2)*(1-x^3*A(x)^2)] + ...
(3) A(x) = 1 + x*A(x)^2/(1-x*A(x)^2) + x^2*A(x)^2/((1-x*A(x)^2)*(1-x^2*A(x)^2)) + x^3*A(x)^2/((1-x*A(x)^2)*(1-x^2*A(x)^2)*(1-x^3*A(x)^2)) + ...

Links

Crossrefs

Cf. A196151, A145268, A145267, A206639, A206637.

Programs

  • Mathematica
    nmax = 30; A[] = 0; Do[A[x] = 1/Product[1 - x^k*A[x]^2, {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Sep 28 2023 *)
(* Calculation of constants {d,c}: *) {1/r, Sqrt[r*s^3 * (s^2 - 1) * Derivative[0, 1][QPochhammer][s^2, r] / (8*Pi*((s^2 - 1)^2*(QPolyGamma[1, 2*Log[s]/Log[r], r] / Log[r]^2) - s^2))]} /. FindRoot[{(1 - s^2)/QPochhammer[s^2, r] == s, 1/2 + s^2/(1 - s^2) == (Log[1 - r] + QPolyGamma[0, 2*Log[s]/Log[r], r]) / Log[r]}, {r, 1/8}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Sep 28 2023 *)
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1/prod(k=1, n, (1-x^k*A^2+x*O(x^n)))); polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*A^(2*m)/prod(k=1, m, (1-x^k+x*O(x^n))))); polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, sqrtint(n+1), x^(m^2)*A^(2*m)/prod(k=1, m, (1-x^k)*(1-x^k*A^2+x*O(x^n))))); polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*A^2/prod(k=1, m, (1-x^k*A^2+x*O(x^n))))); polcoeff(A, n)}

Formula

G.f. satisfies:
(1) A(x) = 1 + Sum_{n>=1} x^n*A(x)^(2*n) / Product_{k=1..n} (1-x^k) due to an identity of Euler.
(2) A(x) = 1 + Sum_{n>=1} x^(n^2)*A(x)^(2*n)/[Product_{k=1..n} (1-x^k)*(1-x^k*A(x)^2)] due to Cauchy's identity.
(3) A(x) = 1 + Sum_{n>=1} x^n*A(x)^2 / Product_{k=1..n} (1 - x^k*A(x)^2).
a(n) ~ c * d^n / n^(3/2), where d = 7.4702934491577480082... and c = 0.270144986991156076... - Vaclav Kotesovec, Aug 12 2021
Radius of convergence r = 0.1338635499135240586... = 1/d and A(r) = 1.5228379370493260575... satisfy A(r) = 1 / sqrt( Sum_{n>=1} 2*r^n/(1 - r^n*A(r)^2) ) and A(r) = 1 / Product_{n>=1} (1 - r^n*A(r)^2). - Paul D. Hanna, Mar 02 2024

A206638 G.f. satisfies: A(x) = Sum_{n>=0} 3^n*A(x)^n * x^(n^2) / Product_{k=1..n} (1 - 3*x^k)*(1 - x^k*A(x)).

Original entry on oeis.org

1, 3, 21, 147, 1074, 8076, 62454, 494292, 3990378, 32756142, 272715870, 2297982828, 19563641319, 168036314862, 1454458825605, 12674387617266, 111104771086812, 979101922849230, 8668964794053837, 77080072176742422, 687976906966730076, 6161811541538326680
Offset: 0

Views

Author

Paul D. Hanna, Feb 10 2012

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 21*x^2 + 147*x^3 + 1074*x^4 + 8076*x^5 +...
where the g.f. satisfies:
(0) A(x) = 1 + 3*x*A(x)/((1-3*x)*(1-x*A(x))) + 9*x^4*A(x)^2/((1-3*x)*(1-3*x^2)*(1-x*A(x))*(1-x^2*A(x))) + 27*x^9*A(x)^3/((1-3*x)*(1-3*x^2)*(1-3*x^3)*(1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))) +...
(1) A(x) = 1 + 3*x*A(x)/(1-3*x) + 3*x^2*A(x)^2/((1-3*x)*(1-3*x^2)) + 3*x^3*A(x)^3/((1-3*x)*(1-3*x^2)*(1-3*x^3)) +...
(2) A(x) = 1 + 3*x*A(x)/(1-x*A(x)) + 9*x^2*A(x)/((1-x*A(x))*(1-x^2*A(x))) + 27*x^3*A(x)/((1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))) +...

Crossrefs

Cf. A145268, A196150, A206637.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, sqrtint(n+1), x^(m^2)*3^m*A^m/prod(k=1, m, (1-3*x^k)*(1-x^k*A+x*O(x^n))))); polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, 3*x^m*A^m/prod(k=1, m, (1-3*x^k+x*O(x^n))))); polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, 3^m*x^m*A/prod(k=1, m, (1-x^k*A+x*O(x^n))))); polcoeff(A, n)}
for(n=0,35,print1(a(n),", "))

Formula

G.f. satisfies the identities:
(1) A(x) = 1 + Sum_{n>=1} 3*x^n*A(x)^n / Product_{k=1..n} (1 - 3*x^k).
(2) A(x) = 1 + Sum_{n>=1} 3^n*x^n*A(x) / Product_{k=1..n} (1 - x^k*A(x)).

A206639 G.f. A(x) satisfies A(x) = Sum_{n>=0} x^(n^2) * A(x)^(2*n) / Product_{k=1..n} (1 - x^k*A(x))^2.

Original entry on oeis.org

1, 1, 4, 18, 91, 489, 2751, 15985, 95218, 578324, 3568084, 22299964, 140885754, 898292262, 5772951668, 37355908797, 243184468271, 1591567315702, 10465836784159, 69114490893596, 458171948148640, 3047865264442504, 20339282134624054, 136122586785459512
Offset: 0

Views

Author

Paul D. Hanna, Feb 11 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 18*x^3 + 91*x^4 + 489*x^5 + 2751*x^6 +...
where the g.f. satisfies:
(0) A(x) = 1 + x*A(x)^2/(1-x*A(x))^2 + x^4*A(x)^4/((1-x*A(x))^2*(1-x^2*A(x))^2) + x^9*A(x)^6/((1-x*A(x))^2*(1-x^2*A(x))^2*(1-x^3*A(x))^2) +...
(1) A(x) = 1 + x*A(x)^2/(1-x*A(x)) + x^2*A(x)^3/((1-x*A(x))*(1-x^2*A(x))) + x^3*A(x)^4/((1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))) +...

Crossrefs

Cf. A196150, A145268, A206637.

Programs

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

Formula

G.f. satisfies the identities:
(1) A(x) = 1 + Sum_{n>=1} x^n*A(x)^(n+1) / Product_{k=1..n} (1 - x^k*A(x)).
(2) A(x) = 1/(1 - Sum_{n>=1} x^n*A(x)^n / Product_{k=1..n} (1 - x^k*A(x)) ).
Showing 1-3 of 3 results.

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