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A298300
Analog of Motzkin numbers for Coxeter type D.
2
1, 4, 11, 31, 87, 246, 699, 1996, 5723, 16468, 47533, 137567, 399073, 1160082, 3378483, 9855207, 28790403, 84218052, 246651729, 723165765, 2122391109, 6234634266, 18330019029, 53932825926, 158802303429, 467898288676, 1379485436579, 4069450219561
OFFSET
2,2
FORMULA
a(n) = A002426(n-1) + A290380(n) (the latter being extended by A290380(2)=0).
Conjectural algebraic equation: 3*t+2+(3*t^2+5*t-2)*f(t)+(3*t^3-t^2)*f(t)^2 = 0.
From Peter Luschny, Jan 23 2018: (Start)
a(n) = hypergeom([(1-n)/2,1-n/2],[1],4)+(n-2)*hypergeom([1-n/2,3/2-n/2],[2],4).
a(n) = G(n-1,1-n,-1/2) + G(n-2,1-n,-1/2)*(n-2)/(n-1) where G(n,a,x) denotes the n-th Gegenbauer polynomial. (End)
D-finite with recurrence +2*n*a(n) +(-7*n+6)*a(n-1) +9*(n-4)*a(n-3)=0. - R. J. Mathar, Jul 27 2022
MAPLE
A298300 := proc(n)
hypergeom([(1-n)/2, 1-n/2], [1], 4)+(n-2)*hypergeom([1-n/2, 3/2-n/2], [2], 4);
simplify(%) ;
end proc:
seq(A298300(n), n=2..40) ; # R. J. Mathar, Jul 27 2022
MATHEMATICA
b[n_] := Hypergeometric2F1[(1 - n)/2, 1 - n/2, 1, 4];
c[n_] := (n-2) Hypergeometric2F1[1 - n/2, 3/2 - n/2, 2, 4];
Table[b[n] + c[n], {n, 2, 29}] (* Peter Luschny, Jan 23 2018 *)
PROG
(SageMath)
def a(n):
return (sum(ZZ(n - 2) / i * binomial(2 * i - 2, i - 1) *
binomial(n - 2, 2 * i - 2)
for i in range(1, floor(n / 2) + 1)) +
sum(binomial(n - 1, k) * binomial(n - 1 - k, k)
for k in range(floor((n - 1) / 2) + 1)))
CROSSREFS
Cf. A001006 (type A), A002426 (type B), A290380.
Sequence in context: A165993 A192312 A004080 * A027115 A077995 A276293
KEYWORD
nonn
AUTHOR
F. Chapoton, Jan 16 2018
STATUS
approved