A372018 G.f. A(x) satisfies A(x) = ( 1 + 4*x*A(x)/(1 - x*A(x)) )^(1/2).
1, 2, 4, 10, 30, 98, 336, 1194, 4360, 16258, 61644, 236938, 921102, 3615330, 14307312, 57024426, 228701646, 922283522, 3737497980, 15212318730, 62160993642, 254909413218, 1048717979424, 4327273358250, 17903826642780, 74260741616514, 308724721176676
Offset: 0
Keywords
Programs
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Maple
A372018 := proc(n) add(4^k*binomial((n+1)/2,k)*binomial(n-1,k-1),k=0..n) ; %/(n+1) ; end proc: seq(A372018(n),n=0..60) ; # R. J. Mathar, Apr 22 2024
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PARI
a(n) = sum(k=0, n, 4^k*binomial(n/2+1/2, k)*binomial(n-1, n-k))/(n+1);
Formula
a(n) = (1/(n+1)) * Sum_{k=0..n} 4^k * binomial(n/2+1/2,k) * binomial(n-1,n-k).
D-finite with recurrence n*(n+1)*(n-2)*a(n) -6*(n-2)*(3*n^2-6*n+1)*a(n-2) -27*n*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Apr 22 2024
Conjecture: a(2n+1) = 2*A371364(). - R. J. Mathar, Apr 22 2024