A365244 G.f. satisfies A(x) = 1 + x*A(x)/(1 - x^2*A(x)^3).
1, 1, 1, 2, 6, 17, 48, 144, 449, 1422, 4568, 14893, 49139, 163665, 549570, 1858754, 6326343, 21651064, 74462327, 257219221, 892047965, 3104749126, 10841192392, 37967942203, 133333407639, 469405472729, 1656383420850, 5857371543403, 20754268304707
Offset: 0
Keywords
Programs
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Maple
A365244 := proc(n) add( binomial(n-k-1,k)*binomial(n+k+1,n-2*k)/(n+k+1),k=0..floor(n/2)) ; end proc: seq(A365244(n),n=0..80); # R. J. Mathar, Aug 29 2023
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Mathematica
nmax = 28; A[_] = 1; Do[A[x_] = 1 + x*A[x]/(1 - x^2*A[x]^3) + O[x]^(nmax+1) // Normal, {nmax}]; CoefficientList[A[x], x] (* Jean-François Alcover, Oct 25 2023 *) -
PARI
a(n) = sum(k=0, n\2, binomial(n-k-1, k)*binomial(n+k+1, n-2*k)/(n+k+1));
Formula
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k-1,k) * binomial(n+k+1,n-2*k)/(n+k+1).
D-finite with recurrence -9*n*(3*n-5) *(3*n+2) *(15657757169*n -38967750523)*a(n) +3*(1246945698477*n^4 -4744568003544*n^3 +3294337649527*n^2 +2214578323972*n -1078893934272) *a(n-1) +6*(98125454565*n^4 -4049050969593*n^3 +21710764341344*n^2 -39026642938410*n +22772957131188) *a(n-2) +6*(1426531749264*n^4 -6603349282173*n^3 -4098111856085*n^2 +51689999346882*n -56245738276010) *a(n-3) +6*(2322713957130*n^4 -32736762801117*n^3 +166244031312630*n^2 -356896536324983*n +268070043432100) *a(n-4) -6*(n-5) *(2*n-9) *(613164767527*n^2 -4657829502565*n +8148618486058) *a(n-5) +2*(n-6) *(2*n-11) *(271184324539*n^2 -2272760427224*n +4256723647917) *a(n-6) -4*(6162243349*n -17166617798) *(2*n-13)*(n-6) *(n-7)*a(n-7)=0. - R. J. Mathar, Aug 29 2023