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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. 0
1, 1, 3, 17, 125, 1055, 9755, 96353, 1000529, 10805045, 120429581, 1377565711, 16103628975, 191753782563, 2319792221739, 28453553549889, 353240252092329, 4432432002187105, 56149270333672577, 717376886229388757, 9236054560816552341, 119742311958753198301, 1562281609210280404333 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
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
Sequence in context: A330803 A267736 A129115 * A093460 A187613 A179300
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 08 2014
STATUS
approved

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Last modified February 22 02:30 EST 2024. Contains 370239 sequences. (Running on oeis4.)