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A364762 G.f. satisfies A(x) = 1 / (1 + x*(1 + x*A(x))^4).

%I A364762 #15 Oct 25 2023 09:28:49
%S A364762 1,-1,-3,5,29,-42,-349,384,4705,-3307,-67530,19392,1006479,140594,
%T A364762 -15356600,-8897336,237691865,246737931,-3708348277,-5655844305,
%U A364762 58027927950,119178376245,-906834380800,-2396063640645,14094956420555,46748815762429,-216921227330074
%N A364762 G.f. satisfies A(x) = 1 / (1 + x*(1 + x*A(x))^4).
%F A364762 a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(n+1,k) * binomial(4*k,n-k).
%F A364762 D-finite with recurrence -128*(2042050230119155915444*n -5429570252206459161379) *(4*n-1) *(2*n-1) *(4*n+5)*(n+1)*a(n) -32*(4*n+1) *(65345607363812989294208*n^4 -820413627047532146653920*n^3 +2763432436839800464356384*n^2 -2486292999757610914452558*n +135739256305161479034475)*a(n-1) +8*(25399908186485156187685696*n^5 -410520476995365688927070656*n^4 +1936659722310862831127714364*n^3 -3768407393014943668311778228*n^2 +3142503008733048326160670659*n -930530980753641871942147310)*a(n-2) +8*(60470966262814634790914176*n^5 -790603842787360719473560256*n^4 +4054580849567079779066278696*n^3 -10338573837841237994265694136*n^2 +13166278761352336081219763480*n-6688902526339456521313590345) *a(n-3) +8*(n-3) *(889289782697551916285417600*n^4 -9728080580320392023685701600*n^3 +38817722034818423528174576784*n^2 -65825556954123813925639696248*n +38902971323615633483566471005) *a(n-4) +8*(n-3) *(n-4)*(2097428215364339549126292456*n^3 -18251533871703386700639649538*n^2 +48220538425138794332555786686*n -34290240482247293887531480365) *a(n-5) +32*(n-3) *(n-4)*(n-5) *(468447167338832041688699407*n^2 -2522399644542316934351318150*n +2141819455274094692083493160) *a(n-6) +3381*(1381857597101978034513050*n -1395473045717316216699133) *(n-3)*(n-4) *(n-5)*(n-6)*a(n-7)=0. - _R. J. Mathar_, Aug 10 2023
%p A364762 A364762 := proc(n)
%p A364762     add( (-1)^k*binomial(n+1,k) * binomial(4*k,n-k),k=0..n) ;
%p A364762     %/(n+1) ;
%p A364762 end proc:
%p A364762 seq(A364762(n),n=0..80); # _R. J. Mathar_, Aug 10 2023
%t A364762 nmax = 26; A[_] = 1;
%t A364762 Do[A[x_] = 1/(1 + x*(1 + x*A[x])^4) + O[x]^(nmax+1) // Normal, {nmax}];
%t A364762 CoefficientList[A[x], x] (* _Jean-François Alcover_, Oct 25 2023 *)
%o A364762 (PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n+1, k)*binomial(4*k, n-k))/(n+1);
%Y A364762 Cf. A007440, A364760, A364761, A364763.
%Y A364762 Cf. A364743.
%K A364762 sign
%O A364762 0,3
%A A364762 _Seiichi Manyama_, Aug 05 2023