A100248 - OEIS

%I #13 Oct 09 2020 03:42:34

%S 1,2,10,79,777,8606,102512,1282129,16605538,220781427,2995985345,

%T 41325515589,577713950666,8166924383923,116550061698966,

%U 1676836298476274,24295472856858786,354190017808427947,5191706917095917442,76469028773023897070,1131207622704483680933,16799374652884761512521

%N Row sums of the slanted Catalan convolution table A100247.

%F a(n) = Sum_{k=0..2n} C(n+2*k-[k/2], k)*(n-[k/2])/(n+2*k-[k/2]).

%F G.f. A(x) satisfies: A(x^2) = ((1+x)/(2 - x*(1-sqrt(1 - 4*x))) - (1-x)/(2 + x*(1-sqrt(1 + 4*x))))/x.

%F a(n) ~ 5 * 2^(4*n + 1/2) / (49*sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Oct 09 2020

%p A100248 := proc(n)

%p     if n = 0 then

%p         1;

%p     else

%p         add(binomial(n+2*k-floor(k/2),k) * (n-floor(k/2)) / (n+2*k-floor(k/2)), k=0..2*n) ;

%p     fi;

%p end proc: # _R. J. Mathar_, May 06 2016

%t CoefficientList[Series[(1 + Sqrt[x])/(2*Sqrt[x] + (-1 + Sqrt[1 - 4*Sqrt[x]])*x) + (1 - Sqrt[x])/(-2*Sqrt[x] + (-1 + Sqrt[1 + 4*Sqrt[x]])*x), {x, 0, 25}], x] (* _Vaclav Kotesovec_, Oct 09 2020 *)

%o (PARI) {a(n)=sum(k=0,2*n,polcoeff(((1-sqrt(1-4*z+z^2*O(z^k)))/(2*z))^(n-k\2),k,z))}

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

%o (PARI) {a(n)=if(n==0,1,sum(k=0,2*n,binomial(n+2*k-(k\2),k)*(n-(k\2))/(n+2*k-(k\2))))}

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

%Y Cf. A100247.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Nov 09 2004