Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
%I #18 Apr 10 2017 22:56:36
%S 1,5,14,45,164,662,2862,12957,60590,290218,1416216,7014714,35172968,
%T 178180968,910542696,4688140189,24296295636,126640986410,663465473910,
%U 3491674292814,18450954171742,97859178176632,520755520521982,2779633126026210
%N a(n) = Sum_{k=0..n}((binomial(2*k,k)*Sum_{i=0..n-k}(binomial(k+1,n-k-i)*binomial(k+i,k)))/(k+1)^2)*(n+1).
%H G. C. Greubel, <a href="/A270661/b270661.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: C((x^2+x)/(1-x))*((x^2-2*x-1)/(x^2-1)*(1+x)/(1-x)), where C(x) is g.f. of Catalan numbers (A000108).
%F a(n) ~ sqrt(55473 + 4469*sqrt(41)) * (5 + sqrt(41))^n / (sqrt(Pi) * n^(3/2) * 2^(n+7)). - _Vaclav Kotesovec_, Mar 21 2016
%F Conjecture: (n+1)*a(n) -4* n*a(n-1) +(-13*n+33)*a(n-2) +2*(8*n-17)*a(n-3) +(19*n-69)*a(n-4) +4*(-4*n+15)*a(n-5) +7*(-n+5)*a(n-6) +2*(2*n-13)*a(n-7)=0. - _R. J. Mathar_, Jun 07 2016
%F Conjecture: (n+1)*(n^2-5*n+3)*a(n) -2*n*(3*n^2-16*n+14)*a(n-1) +2*(n+4)*(n-3)*a(n-2) +2*(5*n^3-31*n^2+39*n+9)*a(n-3) +(-n^3+2*n^2+21)*a(n-4) -2*(2*n-9)*(n^2-3*n-1)*a(n-5)=0. - _R. J. Mathar_, Jun 07 2016
%p A270661 := proc(n)
%p local a,k ;
%p a := 0 ;
%p for k from 0 to n do
%p a := a+binomial(2*k,k)/(k+1)^2*add(binomial(k+1,n-k-i)*binomial(k+i,k),i=0..n-k) ;
%p end do:
%p %*(n+1) ;
%p end proc: # _R. J. Mathar_, Jun 07 2016
%t Table[Sum[(Binomial[2 k, k] Sum[Binomial[k + 1, n - k - i] Binomial[k + i, k], {i, 0, n - k}]/(k + 1)^2) (n + 1), {k, 0, n}], {n, 0, 23}] (* _Michael De Vlieger_, Mar 21 2016 *)
%o (Maxima)
%o C(x):=(1-sqrt(1-4*x))/(2*x);
%o makelist(coeff(taylor(C((x^2+x)/(1-x))/x*((x^2-2*x-1)/(x^2-1)*(1+x)/(1-x)),x,0,10)*x,x,n),n,0,10);
%o a(n):=(sum((binomial(2*k,k)*sum(binomial(k+1,n-k-i)*binomial(k+i,k),i,0,n-k))/(k+1)^2,k,0,n))*(n+1);
%o (PARI) a(n) = sum(k=0, n, (binomial(2*k,k)*sum(i=0, n-k, (binomial(k+1,n-k-i)*binomial(k+i,k)))/(k+1)^2)*(n+1)); \\ _Michel Marcus_, Mar 21 2016
%Y Cf. A000108.
%K nonn
%O 0,2
%A _Vladimir Kruchinin_, Mar 20 2016