A379021 Expansion of (1/x) * Series_Reversion( x * ((1 - x - x^2)/(1 + x))^2 ).
1, 4, 26, 206, 1813, 17032, 167287, 1697044, 17643322, 186997570, 2012973499, 21948003052, 241883091289, 2690117648372, 30153678822007, 340305271736134, 3863616751855069, 44097785533620550, 505692279260755753, 5823592506326814874, 67320958983831426221
Offset: 0
Keywords
Programs
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PARI
a(n) = 2*sum(k=0, n, binomial(2*n+k+2, k)*binomial(2*n+k+2, n-k)/(2*n+k+2));
Formula
G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{k>=1} A379022(k) * x^k/k ).
(2) A(x) = ( (1 + x*A(x)) * (1 + x*A(x)^(3/2)) )^2.
(3) A(x) = B(x)^2 where B(x) is the g.f. of A215654.
a(n) = (1/(n+1)) * [x^n] ( (1 + x)/(1 - x - x^2) )^(2*(n+1)).
a(n) = 2 * Sum_{k=0..n} binomial(2*n+k+2,k) * binomial(2*n+k+2,n-k)/(2*n+k+2) = (1/(n+1)) * Sum_{k=0..n} binomial(2*n+k+1,k) * binomial(2*n+k+2,n-k).