A168358 Self-convolution square of A001246, which is the squares of Catalan numbers.

1, 2, 9, 58, 458, 4120, 40569, 426842, 4723890, 54402904, 646992474, 7900772120, 98642862232, 1254984808672, 16227116787737, 212790354730842, 2824992774357362, 37915366854924952, 513837166842215970

Offset

0,2

Links

Table of n, a(n) for n=0..18.

Formula

G.f.: A(x) = (1/x)*Series_Reversion(x/G(x)^2) where G(x) = g.f. of A006664, which is the number of irreducible systems of meanders.

G.f.: A(x) = G(x*A(x))^2 where A(x/G(x)^2) = G(x)^2 where G(x) = g.f. of A006664.

From Vaclav Kotesovec, Mar 10 2018: (Start)

Recurrence: (n+1)^2*(n+2)^3*(4*n^2 - 5*n - 3)*a(n) = 4*(n+1)^2*(48*n^5 - 12*n^4 - 136*n^3 + 15*n^2 + 49*n - 30)*a(n-1) - 32*(96*n^7 - 312*n^6 + 104*n^5 + 580*n^4 - 630*n^3 + 80*n^2 + 91*n - 12)*a(n-2) + 1024*(n-2)^3*(2*n - 3)^2*(4*n^2 + 3*n - 4)*a(n-3).

a(n) ~ (4/Pi - 1) * 2^(4*n + 3) / (Pi*n^3). (End)

Example

G.f.: A(x) = 1 + 2*x + 9*x^2 + 58*x^3 + 458*x^4 + 4120*x^5 +...
A(x)^(1/2) = 1 + x + 4*x^2 + 25*x^3 + 196*x^4 + 1764*x^5 + 17424*x^6 +...+ A001246(n)*x^n +...
A(x) satisfies: A(x/G(x)^2) = G(x)^2 where G(x) = g.f. of A006664:
G(x) = 1 + x + 2*x^2 + 8*x^3 + 46*x^4 + 322*x^5 + 2546*x^6 +...+ A006664(n)*x^n +...
G(x)^2 = 1 + 2*x + 5*x^2 + 20*x^3 + 112*x^4 + 768*x^5 + 5984*x^6 +...+ A168357(n)*x^n +...

Mathematica

Table[Sum[CatalanNumber[k]^2 * CatalanNumber[n-k]^2, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 10 2018 *)

Prog

(PARI) {a(n)=local(C_2=vector(n+1, m, (binomial(2*m-2, m-1)/m)^2)); polcoeff(Ser(C_2)^2, n)}

Crossrefs

Cf. A168357, A168344, A001246, A006664, A000108.

Sequence in context: A224127 A366401 A116867 * A132608 A361598 A247329

Adjacent sequences: A168355 A168356 A168357 * A168359 A168360 A168361

Keyword

nonn

Author

Paul D. Hanna, Nov 23 2009

Status

approved