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A348479
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Number of interval posets of permutations with n minimal elements.
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3
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1, 1, 3, 12, 52, 240, 1160, 5795, 29681, 155025, 822563, 4421458, 24025518, 131759106, 728330062, 4053823980, 22699853940, 127790656040, 722835069984, 4106096464006, 23414579166050, 133984343279790, 769124367124594, 4427878983496972, 25559244203741228
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OFFSET
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1,3
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LINKS
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FORMULA
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a(n) = (1/n) * Sum_{i=1..(n-1)} Sum_{k=0..Min(i,(n-i-1)/2)} binomial(n+i-1,i)* binomial(i,k)*binomial(n-2k-2,i-1) if n>1. Proved in M. Bouvel, L. Cioni, B. Izart (Theorem 18).
G.f. A(z) = Sum_{n>=0} a(n)*z^n satisfies the equation A(z) = z + (A(z)^2 + A(z)^4)/(1-A(z)). Proved in M. Bouvel, L. Cioni, B. Izart (Equation (1) page 14).
Asymptotic behavior of a(n) is c*n^(-3/2)*r^n with c approximately 0.0622 and r approximately 6.1403. Proved in M. Bouvel, L. Cioni, B. Izart (Theorem 19).
D-finite with recurrence 177*n*(n-1)*(n-2) *(1884*n-6797)*a(n) -(n-1) *(n-2) *(2079652*n^2-10492117*n+10802220) *a(n-1) +6*(n-2) *(98404*n^3-611787*n^2+893503*n+124240) *a(n-2) +2*(-1206916*n^4+13262653*n^3-52943063*n^2+90096428*n-54243072) *a(n-3) +(-16564*n^4+1171171*n^3-12487565*n^2+47878166*n-62441016) *a(n-4) +3 *(3*n-14) *(n-5) *(388*n-1861) *(3*n-16)*a(n-5)=0. - R. J. Mathar, Nov 04 2021
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MATHEMATICA
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Join[{1}, Table[Sum[Sum[Binomial[n+i-1, i]Binomial[i, k]Binomial[n-2k-2, i-1], {k, 0, Min[i, (n-i-1)/2]}], {i, n-1}]/n, {n, 2, 25}]] (* Stefano Spezia, Oct 23 2021 *)
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PROG
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(PARI) a(n) = if (n==1, 1, (1/n) * sum(i=1, n-1, sum(k=0, min(i, (n-i-1)/2), binomial(n+i-1, i)* binomial(i, k)*binomial(n-2*k-2, i-1)))); \\ Michel Marcus, Oct 21 2021
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CROSSREFS
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For interval posets which are in addition trees, see A054515.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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