OFFSET
0,2
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Peter J. Taylor, Table of n, a(n) for n = 0..500 (terms 0..50 from Remigiusz Suwalski)
P. N. Rathie, The enumeration of Hamiltonian polygons in rooted planar triangulations, Discrete Math., 6 (1973), 163-168.
FORMULA
r(n) = (binomial(2*n, n) / (n + 1))^2.
B(s, m) = sum((m! / m_1! ... m_s!) * r(1)^{m_1} ... r(s)^{m_s}) where the sum is over all partitions of s such that s = m_1 + 2*m_2 + ... + s*m_s and m = m_1 + m_2 + ... + m_s.
A(n, s) = Sum_{m=1..s} binomial(n, m) * B(s, m).
p(n, k) = k * (2*n + 2*k - 4)! * (2*n + k - 1)! / ((n + k - 1)! * (n + k - 2)! * n! * (n + k)!).
f(n, k) = p(n, k) - Sum_{s=0..n-1} f(s, k) * A(k+s, n-s).
a(n) = f(n, 3). - Sean A. Irvine, Feb 02 2015
MATHEMATICA
functiony[l_] :=
If[Range[Length[l]].l > Length[l], {}, len = Length[l];
Select[Permutations[l], #.Range[len] == len &]]
functionb[s_, m_] := Module[{l = 0},
If[m + s == 0, 1,
If[m s == 0, 0,
If[m >= s,
If[m > s, 0, 1],
If[m == 1, CatalanNumber[s]^2,
If[s - m == 1, 4 m,
l =
Flatten[Map[functiony, IntegerPartitions[s + m, {s}] - 1], 1];
Map[Times @@ # &,
Map[Map[r, Range[1, s]]^# &,
l]].(Map[Times @@ # &, Map[Factorial, l]])^(-1)*m!]
]
]
]
]
]
a[n_, s_] := Sum[Binomial[n, m] b[s, m], {m, 1, s}]
b[s_, m_] :=
If[s + m > 0, table1[[s + 1, m + 1]], If[s + m == 0, 1, 0]]
f[n_, k_] :=
k (2 n + 2 k -
4)! (2 n + k - 1)!/((n + k - 1)! (n + k - 2)! n! (n + k)!) -
Sum[table2[[s + 1]] a[k + s, n - s], {s, 0, n - 1}]
r[n_] := (Binomial[2 n, n])^2/(n^2 + 2 n + 1)
answer[n_] := f[n, 3]
index = 24;
table1 = Table[functionb[s, m], {s, 0, index}, {m, 0, index}];
table2 = Range[index];
For[i = 2, i <= index, i++, table2[[i]] = f[i - 1, 3]];
f[index - 1, 3]
(* Xesda Gonia, Dec 29 2015 *)
PROG
(C#) See Taylor link
(PARI)
P(n, k) = k*(2*n+2*k-4)!*(2*n+k-1)!/((n+k-1)!*(n+k-2)!*n!*(n+k)!);
F(K, N=23) = {
my(x='x + O('x^(K+1)), t='t + O('t^(N+1)),
r='t*Ser(vector(N, n, sqr(binomial(2*n, n)/(n+1))), 't),
p=x^3*Ser(apply(k->Ser(vector(N, n, P(n-1, k)), 't), [3..K])),
s=serreverse(t*(1+r)), f=subst(subst(p, 't, s), 'x, 'x*s/'t));
Vec(polcoeff(f, K));
};
F(3) \\ Gheorghe Coserea, Aug 18 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms and title clarified by Sean A. Irvine, Feb 02 2015
Three more terms from Xesda Gonia, Dec 29 2015
STATUS
approved