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.

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A277309 G.f. satisfies: A(x - 5*A(x)^2) = x - 3*A(x)^2.

Original entry on oeis.org

1, 2, 28, 570, 14284, 410604, 13046728, 448252682, 16417945620, 634848045084, 25737059674104, 1088311917852828, 47813839403065432, 2175881570186952520, 102316326149365110320, 4961686220242926811690, 247733650768933667153660, 12718117037478356041212500, 670565414769224589112024760, 36274908884974158393988101900, 2011581759381610503724213971960
Offset: 1

Views

Author

Paul D. Hanna, Oct 09 2016

Keywords

Examples

			G.f.: A(x) = x + 2*x^2 + 28*x^3 + 570*x^4 + 14284*x^5 + 410604*x^6 + 13046728*x^7 + 448252682*x^8 + 16417945620*x^9 + 634848045084*x^10 +...

Links

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.
Cf. A277300, A277301, A277302, A277303, A277304, A277305, A277306, A277307, A277308.
Cf. A276364.

Programs

  • PARI
    {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x-5*F^2) + 3*F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + 2 * A( 5*A(x)/2 - 3*x/2 )^2.
(2) A(x) = 3*x/5 + 2/5 * Series_Reversion(x - 5*A(x)^2).
(3) R(x) = 5*x/3 - 2/3 * Series_Reversion(x - 3*A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/2 - R(x)/2 ) ) = 5*x/2 - 3*R(x)/2, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k) * 5^k * 2^(n-k-1).

A276366 G.f. A(x) satisfies: A(x - A(x)^3) = x + A(x)^2.

Original entry on oeis.org

1, 1, 3, 12, 57, 300, 1697, 10126, 62991, 405247, 2680901, 18160444, 125562250, 883868590, 6321838520, 45869309028, 337167193262, 2508018933431, 18861358215299, 143293615189089, 1098997404472941, 8504070741463729, 66358269984208701, 521923129718567918, 4136089275165532156, 33013640650845937124
Offset: 1

Views

Author

Paul D. Hanna, Sep 01 2016

Keywords

Examples

			G.f.: A(x) = x + x^2 + 3*x^3 + 12*x^4 + 57*x^5 + 300*x^6 + 1697*x^7 + 10126*x^8 + 62991*x^9 + 405247*x^10 + 2680901*x^11 + 18160444*x^12 +...
such that A(x - A(x)^3) = x + A(x)^2.
RELATED SERIES.
A(x - A(x)^3) = x + x^2 + 2*x^3 + 7*x^4 + 30*x^5 + 147*x^6 + 786*x^7 + 4480*x^8 + 26814*x^9 + 166865*x^10 + 1072160*x^11 + 7076724*x^12 +...
which equals x + A(x)^2.

Links

Crossrefs

Cf. A153851, A275765, A276364.

Programs

  • PARI
    {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x-F^3) - F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies: A'(x - A(x)^3) = (1 + 2*A'(x)*A(x)) / (1 - 3*A'(x)*A(x)^2).
Previous Showing 11-12 of 12 results.

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