A364762 - OEIS

A364762

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

1, -1, -3, 5, 29, -42, -349, 384, 4705, -3307, -67530, 19392, 1006479, 140594, -15356600, -8897336, 237691865, 246737931, -3708348277, -5655844305, 58027927950, 119178376245, -906834380800, -2396063640645, 14094956420555, 46748815762429, -216921227330074

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OFFSET

0,3

LINKS

Table of n, a(n) for n=0..26.

FORMULA

a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(n+1,k) * binomial(4*k,n-k).

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

MAPLE

A364762 := proc(n)

add( (-1)^k*binomial(n+1, k) * binomial(4*k, n-k), k=0..n) ;

%/(n+1) ;

end proc:

seq(A364762(n), n=0..80); # R. J. Mathar, Aug 10 2023

MATHEMATICA

nmax = 26; A[_] = 1;

Do[A[x_] = 1/(1 + x*(1 + x*A[x])^4) + O[x]^(nmax+1) // Normal, {nmax}];

CoefficientList[A[x], x] (* Jean-François Alcover, Oct 25 2023 *)

PROG

(PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n+1, k)*binomial(4*k, n-k))/(n+1);