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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

%I #11 Sep 16 2014 07:24:10

%S 1,1,3,17,125,1055,9755,96353,1000529,10805045,120429581,1377565711,

%T 16103628975,191753782563,2319792221739,28453553549889,

%U 353240252092329,4432432002187105,56149270333672577,717376886229388757,9236054560816552341,119742311958753198301,1562281609210280404333

%N 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.

%F 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.

%F 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).

%F 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

%F a(n) ~ 12*(7-4*sqrt(3)) * (8+4*sqrt(3))^n / (Pi * n^3). - _Vaclav Kotesovec_, Sep 16 2014

%e G.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 125*x^4 + 1055*x^5 + 9755*x^6 +...

%e The g.f. A(x) = D(x-x^2) satisfies A(x*C(x)) = D(x), where

%e 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 +...

%e 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 +...

%e The squares of the Catalan numbers can be generated from this sequence by using the following summation rule.

%e Drop the first term, take partial sums, and repeat, like so:

%e 1, 1, 3, 17, 125, 1055, 9755, 96353, 1000529, 10805045, ...;

%e .. 1, 4, 21, 146, 1201, 10956, 107309, 1107838, 11912883, ...;

%e ..... 4, 25, 171, 1372, 12328, 119637, 1227475, 13140358, ...;

%e ........ 25, 196, 1568, 13896, 133533, 1361008, 14501366, ...;

%e ............ 196, 1764, 15660, 149193, 1510201, 16011567, ...;

%e ................. 1764, 17424, 166617, 1676818, 17688385, ...;

%e ....................... 17424, 184041, 1860859, 19549244, ...;

%e .............................. 184041, 2044900, 21594144, ...; ...

%t 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 *)

%o (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)}

%o for(n=0,30,print1(a(n),", "))

%o (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)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A000108, A001246.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 08 2014

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