A184360 -id:A184360 - cp's OEIS frontend

A184361 Self-convolution equals A184360.

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

Offset: 0

Paul D. Hanna, Jan 16 2011

nonn

			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 +...

Cf. A184358, A184359, A184360.

{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)}

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).

A184358 a(n) = (n+1)!^2/2^n.

Original entry on oeis.org

1, 2, 9, 72, 900, 16200, 396900, 12700800, 514382400, 25719120000, 1556006760000, 112032486720000, 9466745127840000, 927741022528320000, 104370865034436000000, 13359470724407808000000, 1930443519676928256000000, 312731850187662377472000000

Offset: 0

Paul D. Hanna, Jan 16 2011

nonn

Self-convolution of A184359.

			G.f.: A(x) = 1 + 2*x + 9*x^2 + 72*x^3 + 900*x^4 + 16200*x^5 +...
A(x)^(1/2) = 1 + x + 4*x^2 + 32*x^3 + 410*x^4 + 7562*x^5 + 188736*x^6 +...+ A184359(n)*x^n +...

Cf. A184359, A184360, A184361, A006472.

a[n_] := (n + 1)!^2/2^n; Array[a, 20, 0] (* Amiram Eldar, Jun 25 2022 *)

PARI
```
{a(n)=(n+1)!^2/2^n}
```

From Amiram Eldar, Jun 25 2022: (Start)
Sum_{n>=0} 1/a(n) = (BesselI(0, 2*sqrt(2)) - 1)/2.
Sum_{n>=0} (-1)^n/a(n) = (1 - BesselJ(0, 2*sqrt(2)))/2. (End)

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

Paul D. Hanna, Jan 16 2011

nonn

			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 +...

Cf. A184358, A184360, A184361.

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

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.

cp's OEIS Frontend

A184361 Self-convolution equals A184360.

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A184358 a(n) = (n+1)!^2/2^n.

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

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