A228192 - OEIS

%I #4 Aug 15 2013 13:47:57

%S 1,3,321,1462563,252055236609,1569245074591690083,

%T 345299757825442889707393857,2653337188651000290233505189314055363,

%U 706829476163540077094231781323762631545566527489,6496844758902641761809431955916116052361210081093847336070467

%N  a(n) = A001850(n^2), where A001850 forms the central Delannoy numbers.

%F  a(n) = Sum_{k=0..n^2} binomial(n^2, k) * binomial(n^2+k, k).

%F a(n) = [x^(n^2)] 1/sqrt(1 - 6*x + x^2).

%F Equals the logarithmic derivative of A228193, after ignoring the initial term.

%e  L.g.f.: L(x) = 3*x + 321*x^2/2 + 1462563*x^3/3 + 252055236609*x^4/4 +...

%e where exponentiation yields the g.f. of A228193:

%e exp(L(x)) = 1 + 3*x + 165*x^2 + 488007*x^3 + 63015285321*x^4 + 313849204040245803*x^5 +...+ A228193(n)*X^n +...

%o  (PARI) {a(n)=sum(k=0,n^2,binomial(n^2,k)*binomial(n^2+k,k))}

%o for(n=0,20,print1(a(n),", "))

%o (PARI) {A001850(n)=polcoeff(1/sqrt(1 - 6*x + x^2 + x*O(x^n)), n)}

%o {a(n)=A001850(n^2)}

%o for(n=0,20,print1(a(n),", "))

%Y  Cf. A228193, A001850.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 15 2013