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.

A184360 G.f.: A(x) = x/Series_Reversion(x*G(x)) where G(x) = Sum_{n>=0} (n+1)!^2*(x/2)^n.

Original entry on oeis.org

1, 2, 5, 34, 442, 8638, 229467, 7862664, 336468450, 17579403622, 1101881183359, 81669937516066, 7070184169543820, 707266516140720872, 80989516005804384644, 10528134125581145088720, 1542184766049169920609018
Offset: 0

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Author

Paul D. Hanna, Jan 16 2011

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 34*x^3 + 442*x^4 + 8638*x^5 +...
A(x)^(1/2) = 1 + x + 2*x^2 + 15*x^3 + 204*x^4 + 4085*x^5 + 110128*x^6 +...+ A184361(n)*x^n +...
The g.f. of A184358 is G(x) = A(x*G(x)):
G(x) = 1 + 2*x + 9*x^2 + 72*x^3 + 900*x^4 + 16200*x^5 + 396900*x^6 +...+ (n+1)!^2*x^n/2^n +...

Crossrefs

Cf. A184361, A184358, A182958.

Programs

  • PARI
    {a(n)=polcoeff(x/serreverse(x*sum(m=0,n+1,(m+1)!^2*(x/2)^m)+x^2*O(x^n)),n)}

Formula

G.f. satisfies: A(x) = G(x/A(x)) where A(x*G(x)) = G(x) = Sum_{n>=0} (n+1)!^2*(x/2)^n.
G.f. satisfies: [x^n] A(x)^(n+1)/(n+1) = (n+1)!^2/2^n = A184358(n).

A184361 Self-convolution equals A184360.

Original entry on oeis.org

1, 1, 2, 15, 204, 4085, 110128, 3809974, 164121912, 8615474691, 541908913830, 40272139958565, 3493551786163290, 350048185790908410, 40136947555438179728, 5223165612267081234916, 765782709626083599128656
Offset: 0

Views

Author

Paul D. Hanna, Jan 16 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 15*x^3 + 204*x^4 + 4085*x^5 +...
A(x)^2 = 1 + 2*x + 5*x^2 + 34*x^3 + 442*x^4 + 8638*x^5 + 229467*x^6 +...+ A184360(n)*x^n +...

Crossrefs

Cf. A184358, A184359, A184360.

Programs

  • PARI
    {a(n)=local(G=sum(m=0,n,(m+1)!^2*x^m/2^m)+x*O(x^n));polcoeff(sqrt(x/serreverse(x*G)),n)}

Formula

G.f. satisfies: A(x) = G(x/A(x)^2) and A(x*G(x)^2) = G(x) is the g.f. of A184359.
G.f.: A(x) = sqrt(x/Series_Reversion(x*F(x))) where F(x) = Sum_{n>=0} (n+1)!^2*(x/2)^n is the g.f. of A184358.
G.f. satisfies: [x^n] A(x)^(2n+2)/(n+1) = (n+1)!^2/2^n = A184358(n).

A184359 Recurrence: Sum_{n>=0} a(n-k)*a(k) = (n+1)!^2/2^n.

Original entry on oeis.org

1, 1, 4, 32, 410, 7562, 188736, 6118296, 249991926, 12575954918, 764125698224, 55189878377480, 4674557178309388, 458942541226822876, 51705551381013381112, 6626012145599584408536, 958371653002293850802814
Offset: 0

Views

Author

Paul D. Hanna, Jan 16 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 32*x^3 + 410*x^4 + 7562*x^5 +...
A(x)^2 = 1 + 2*x + 9*x^2 + 72*x^3 + 900*x^4 + 16200*x^5 + 396900*x^6 +...+ (n+1)!^2*x^n/2^n +...
The g.f. of A184361 is F(x) = A(x/F(x)^2):
F(x) = 1 + x + 2*x^2 + 15*x^3 + 204*x^4 + 4085*x^5 + 110128*x^6 +...

Crossrefs

Cf. A184358, A184360, A184361.

Programs

  • PARI
    {a(n)=local(G=sum(m=0,n,(m+1)!^2*x^m/2^m)+x*O(x^n));polcoeff(sqrt(G),n)}

Formula

Self-convolution equals A184358.
G.f. satisfies: A(x) = F(x*A(x)^2) where A(x/F(x)^2) = F(x) is the g.f. of A184361.
G.f.: A(x) = sqrt((1/x)*Series_Reversion(x/F(x)^2)) where F(x) is the g.f. of A184361.
Showing 1-3 of 3 results.

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