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A218045
Number of truth tables of bracketed formulas (case 3).
6
0, 0, 1, 2, 9, 46, 262, 1588, 10053, 65686, 439658, 2999116, 20774154, 145726348, 1033125004, 7390626280, 53281906861, 386732675046, 2823690230850, 20725376703324, 152833785130398, 1131770853856100, 8412813651862868
OFFSET
0,4
COMMENTS
Equals the self-convolution of A186997 (up to offset). - Paul D. Hanna, Jul 03 2023
FORMULA
Yildiz gives a g.f.: (2+2*sqrt(1-8*x)-(1+sqrt(1-8*x))*sqrt(2+2*sqrt(1-8*x)+8*x))/8.
a(n+1) = (Sum_{k = 0..n} (Sum_{i=0..n-k} (binomial(k, 2*k+i+1-n)*binomial(k+i-1, i)))*binomial(2*n,k))/n. - Vladimir Kruchinin, Nov 19 2014
G.f. G(x) = A(x)/x satisfies G(x) = x*((G(x)*(G(x)+1))/(1-G(x))+1)^2. - Vladimir Kruchinin, Nov 19 2014
a(n) ~ (2*sqrt(3)-3) * 2^(3*n-3) / (3 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 19 2014
From Paul D. Hanna, Jul 03 2023: (Start)
G.f. A(x) = Series_Reversion( x*(1 + sqrt(1 - 4*x - 4*x^2)) / 2 )^2.
G.f. A(x) = exp( Sum_{n>=1} A288470(n) * x^n/n ), where A288470(n) = Sum_{k=0..n} binomial(n,k) * binomial(2*n,2*k). (End)
EXAMPLE
G.f. A(x) = x^2 + 2*x^3 + 9*x^4 + 46*x^5 + 262*x^6 + 1588*x^7 + 10053*x^8 + 65686*x^9 + 439658*x^10 + ...
MATHEMATICA
CoefficientList[Series[(2+2*Sqrt[1-8*x]-(1+Sqrt[1-8*x])*Sqrt[2+2*Sqrt[1-8*x]+8*x])/8, {x, 0, 20}], x] (* Vaclav Kotesovec, Nov 19 2014 after Yildiz *)
Flatten[{0, 0, Table[Sum[(Sum[Binomial[k, 2*k+i+2-n]*Binomial[k+i-1, i], {i, 0, n-k-1}]*Binomial[2*n-2, k])/(n-1), {k, 0, n-1}], {n, 2, 20}]}] (* Vaclav Kotesovec, Nov 19 2014 after Vladimir Kruchinin *)
PROG
(Maxima)
a(n):=sum((sum(binomial(k, 2*k+i-n)*binomial(k+i-1, i), i, 0, n-k+1))*binomial(2*n+2, k), k, 0, n+1)/(n+1); /* Vladimir Kruchinin, Nov 19 2014 */
(PARI) x='x+O('x^50); concat([0, 0], Vec((2+2*sqrt(1-8*x)-(1+sqrt(1-8*x))*sqrt(2 + 2*sqrt(1-8*x)+8*x))/8)) \\ G. C. Greubel, Apr 01 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 23 2012
STATUS
approved