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.

A238223 Decimal expansion of a constant related to A088716.

Original entry on oeis.org

2, 1, 7, 9, 5, 0, 7, 8, 9, 4, 4, 7, 1, 5, 1, 0, 6, 5, 4, 9, 2, 8, 2, 2, 8, 2, 2, 4, 4, 2, 3, 1, 9, 8, 2, 0, 8, 8, 6, 6, 0, 4, 5, 3, 9, 5, 6, 2, 9, 3, 9, 9, 6, 3, 4, 8, 1, 2, 3, 4, 0, 1, 7, 6, 2, 6, 5, 8, 7, 3, 3, 6, 2, 9, 2, 5, 3, 7, 0, 9, 4, 4, 9, 1, 2, 5, 9, 6, 3, 2, 2, 9, 8, 6, 2, 2, 9, 4, 5, 1, 4, 4, 8, 8, 9, 0
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 21 2014

Keywords

Examples

			0.21795078944715106549282282244231982088...

Links

Crossrefs

Cf. A088716, A088715, A158884, A159606, A182962, A193332, A156326, A338377.

Formula

Equals lim n->infinity A088716(n)/(n!*n^2).

A159607 G.f. satisfies: A(x) = 1 + x*d/dx log(1 + x*A(x)^2).

Original entry on oeis.org

1, 1, 3, 16, 123, 1221, 14724, 207908, 3355803, 60873595, 1225319163, 27097430328, 653052022740, 17036213760892, 478306368143880, 14381009543824236, 461038595072589531, 15699544671941958663, 565927686301436324649
Offset: 0

Views

Author

Paul D. Hanna, May 16 2009

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 123*x^4 + 1221*x^5 +...
A(x)^2 = 1 + 2*x + 7*x^2 + 38*x^3 + 287*x^4 + 2784*x^5 +...
log(1+x*A(x)^2) = x + 3*x^2/2 + 16*x^3/3 + 123*x^4/4 + 1221*x^5/5 +...

Links

Crossrefs

Cf. variants: A159606, A159608.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+x*deriv(log(1+x*Ser(A)^2)+x*O(x^n)));polcoeff(A,n)}

Formula

G.f. satisfies: A(x) = 1 + x*A(x)^2*(2 - A(x)) + 2*x^2*A'(x)*A(x).
a(n) ~ c * n! * 2^n, where c = 0.343014753433948245763329120820010283... - Vaclav Kotesovec, Feb 22 2014

A159608 G.f. satisfies: A(x) = 1 + x*d/dx log(1 + x*A(x)^3).

Original entry on oeis.org

1, 1, 5, 46, 597, 9791, 191876, 4348394, 111561125, 3192096511, 100729014305, 3474750994936, 130094553648612, 5254546985647116, 227771218849108212, 10548385893161367506, 519835256567911242341, 27164324421130818956039
Offset: 0

Views

Author

Paul D. Hanna, May 16 2009

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 46*x^3 + 597*x^4 + 9791*x^5 +...
A(x)^3 = 1 + 3*x + 18*x^2 + 169*x^3 + 2157*x^4 + 34548*x^5 +...
log(1+x*A(x)^3) = x + 5*x^2/2 + 46*x^3/3 + 597*x^4/4 + 9791*x^5/5 +...

Links

Crossrefs

Cf. variants: A159606, A159607.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+x*deriv(log(1+x*Ser(A)^3)+x*O(x^n)));polcoeff(A,n)}

Formula

G.f. satisfies: A(x) = 1 + x*(2 - A(x))*A(x)^3 + 3*x^2*A'(x)*A(x)^2.
a(n) ~ c * 3^n * n! * n^(1/3), where c = 0.242604467523310747298... - Vaclav Kotesovec, Aug 24 2017
Showing 1-3 of 3 results.

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