A246963 G.f. satisfies: A(x) = Sum_{n>=0} A000108(n)^2 * (x-x^2)^n, where A000108(n) = C(2*n,n)/(n+1) is the n-th Catalan number.

1, 1, 3, 17, 125, 1055, 9755, 96353, 1000529, 10805045, 120429581, 1377565711, 16103628975, 191753782563, 2319792221739, 28453553549889, 353240252092329, 4432432002187105, 56149270333672577, 717376886229388757, 9236054560816552341, 119742311958753198301, 1562281609210280404333

Offset

0,3

Links

Table of n, a(n) for n=0..22.

Formula

G.f. A(x) satisfies: A(x*C(x)) = Sum_{n>=0} A000108(n)^2 * x^n, where C(x) = 1 + x*C(x)^2 is the Catalan function.

a(n) = Sum_{k=0..n} (-1)^k * C(n-k,k) * A000108(n-k)^2, where A000108(n) = C(2*n,n)/(n+1).

Recurrence: n*(n+1)^2*a(n) = n*(18*n^2 - 13*n + 5)*a(n-1) - (49*n^3 - 111*n^2 + 20*n + 72)*a(n-2) + 8*(6*n^3 - 22*n^2 + 8*n + 27)*a(n-3) - 16*(n-3)^2*(n+1)*a(n-4). - Vaclav Kotesovec, Sep 16 2014

a(n) ~ 12*(7-4*sqrt(3)) * (8+4*sqrt(3))^n / (Pi * n^3). - Vaclav Kotesovec, Sep 16 2014

Example

G.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 125*x^4 + 1055*x^5 + 9755*x^6 +...
The g.f. A(x) = D(x-x^2) satisfies A(x*C(x)) = D(x), where
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + 1430*x^8 + 4862*x^9 +...+ A000108(n)*x^n +...
D(x) = 1 + x + 4*x^2 + 25*x^3 + 196*x^4 + 1764*x^5 + 17424*x^6 + 184041*x^7 + 2044900*x^8 + 23639044*x^9 +...+ A000108(n)^2*x^n +...
The squares of the Catalan numbers can be generated from this sequence by using the following summation rule.
Drop the first term, take partial sums, and repeat, like so:
1, 1, 3, 17, 125, 1055,  9755,  96353, 1000529, 10805045, ...;
.. 1, 4, 21, 146, 1201, 10956, 107309, 1107838, 11912883, ...;
..... 4, 25, 171, 1372, 12328, 119637, 1227475, 13140358, ...;
........ 25, 196, 1568, 13896, 133533, 1361008, 14501366, ...;
............ 196, 1764, 15660, 149193, 1510201, 16011567, ...;
................. 1764, 17424, 166617, 1676818, 17688385, ...;
....................... 17424, 184041, 1860859, 19549244, ...;
.............................. 184041, 2044900, 21594144, ...; ...

Mathematica

Table[Sum[(-1)^k * Binomial[n-k, k] * Binomial[2*(n-k), n-k]^2 / (n-k+1)^2, {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 16 2014 *)

Prog

(PARI) {a(n)=local(A=1); A=sum(m=0, n, binomial(2*m, m)^2/(m+1)^2 * (x-x^2 +x*O(x^n))^m); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=sum(k=0, n\2, (-1)^k * binomial(n-k, k) * binomial(2*(n-k), n-k)^2/(n-k+1)^2)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A000108, A001246.

Sequence in context: A330803 A267736 A129115 * A093460 A187613 A179300

Adjacent sequences: A246960 A246961 A246962 * A246964 A246965 A246966

Keyword

nonn

Author

Paul D. Hanna, Sep 08 2014

Status

approved