OFFSET
1,2
COMMENTS
a(n) is the maximum number of distinct sets that can be obtained as complete parenthesizations of “S_1 union S_2 intersect S_3 union S_4 intersect S_5 union ... union S_{2*n}”, where n union and n-1 intersection operations alternate, starting with a union, and S_1, S_2, ... , S_{2*n} are sets. - Alexander Burstein, Nov 22 2023
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..950
Emeric Deutsch, Problem 10658, American Math. Monthly, 107, 2000, 368-370.
FORMULA
G.f.: z*A^2, where A is the g.f. of A027307.
a(n) = 2*Sum_{i=0..n-1} (2*n+i-1)!/(i!*(n-i-1)!*(n+i+1)!). - Vladimir Kruchinin, Oct 18 2011
D-finite with recurrence: n*(2*n-1)*a(n) = (28*n^2-65*n+36)*a(n-1) - (64*n^2-323*n+408)*a(n-2) - 3*(n-4)*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 08 2012
a(n) ~ sqrt(45*sqrt(5)-100)*((11+5*sqrt(5))/2)^n/(5*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 08 2012
G.f. A(x) satisfies: A(x) = 1 + x * ( A(x) + sqrt(A(x)) )^2. - Paul D. Hanna, Jun 11 2016
From Peter Bala, May 07 2023: (Start)
n*(2*n-1)*(5*n-9)*a(n) = 2*(55*n^3-209*n^2+255*n-99)*a(n-1) + (n-3)*(2*n-3)*(5*n-4)*a(n-2) with a(1) = 1 and a(2) = 4.
G.f.: A(x) = series reversion of x*(1 - x)^2/(1 + x)^2. (End)
MATHEMATICA
RecurrenceTable[{n*(2*n-1)*a[n] == (28*n^2-65*n+36)*a[n-1] - (64*n^2-323*n+408)*a[n-2] - 3*(n-4)*(2*n-5)*a[n-3], a[1]==1, a[2]==4, a[3]==24}, a, {n, 20}] (* Vaclav Kotesovec, Oct 08 2012 *)
PROG
(Maxima)
a(n):=2*sum((2*n+i-1)!/(i!*(n-i-1)!*(n+i+1)!), i, 0, n-1); \\ Vladimir Kruchinin, Oct 18 2011
(PARI) vector(30, n, 2*sum(k=0, n-1, (2*n+k-1)!/(k!*(n-k-1)!*(n+k+1)!))) \\ Altug Alkan, Oct 06 2015
(PARI) {a(n) = my(A=1); for(i=1, n, A = 1 + x*(A + sqrt(A +x*O(x^n)))^2); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Jun 11 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved