A092255 a(0) = 1; for n > 0, a(n) = b(n) - n*b(n-1), b() = A076177().

1, 1, 3, 10, 23, 66, 222, 561, 1647, 5410, 14318, 42351, 137018, 372191, 1105275, 3537540, 9772767, 29090826, 92364198, 258208671, 769820418, 2429091885, 6850744365, 20447143866, 64200928194, 182303186391, 544550917797, 1702925802766, 4861918919447

Offset: 0

Author

N. J. A. Sloane, Feb 20 2004

Keywords

• nonn

Comments

For n>=1, a(n) mod 2 = A010060(n) the Thue-Morse sequence - Benoit Cloitre, Mar 22 2004

Number of ternary words of length n in which count(0's) <= count(1's) <= count(2's). a(2) = 3: words 12 and 21 with counts (0,1,1) and 22 with counts (0,0,2). - David Scambler, Aug 06 2012

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Column k=3 of A226873. - Alois P. Heinz, Jun 21 2013

Programs

Maple

a:= n-> n! *add(add(1/(k!*j!*(n-k-j)!), j=k..(n-k)/2), k=0..n/3):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 07 2012

Mathematica

CoefficientList[Series[(HypergeometricPFQ[{},{},x]^3 + 3*HypergeometricPFQ[{},{},x]*HypergeometricPFQ[{},{1},x^2] + 2*HypergeometricPFQ[{},{1,1},x^3])/6,{x,0,30}],x]*Range[0,30]! (* Vaclav Kotesovec, Jul 01 2013 *)

PARI

a(n)=sum(i=0,n,sum(j=0,i,sum(k=0,j,if(i+j+k-n,0,n!/i!/j!/k!))))

Formula

a(n) = n!*sum(i+j+k=n, 1/(i!*j!*k!)) (0<=k<=j<=i<=n). - Benoit Cloitre, Mar 22 2004

Recurrence: (n-3)*(n-1)*n^2*(63*n^3-561*n^2+1556*n-1343)*a(n) = (n-1)^2*(315*n^5 - 4065*n^4 + 19720*n^3 - 44240*n^2 + 44790*n - 15768)*a(n-1) - 3*(63*n^4 - 813*n^3 + 3392*n^2 - 5091*n + 2241)*(n-2)^3*a(n-2) + 9*(n-3)*(126*n^5 - 1311*n^4 + 4801*n^3 - 7677*n^2 + 5409*n - 1464)*(n-2)*a(n-3) - 27*(n-3)*(315*n^5 - 4065*n^4 + 19720*n^3 - 44240*n^2 + 44790*n - 15768)*(n-2)*a(n-4) + 81*(n-4)*(n-3)*(63*n^4 - 813*n^3 + 3392*n^2 - 5091*n + 2241)*(n-2)*a(n-5) + 243*(n-5)*(n-4)*(n-3)*(63*n^3-372*n^2+623*n-285)*(n-2)*a(n-6). - Vaclav Kotesovec, Jun 30 2013

a(n) ~ 1/2 * 3^(n-1) * (1 + 3/(2*sqrt(Pi*n/3)) + sqrt(3)*(1+2*cos(2*Pi*n/3))/(Pi*n)). - Vaclav Kotesovec, Mar 07 2014

Extensions

More terms from Benoit Cloitre, Mar 22 2004