A096568 By rows, array T(n,k)=number of compositions of n with first part k and no equal adjacent parts.

1, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 2, 1, 1, 1, 5, 4, 2, 1, 1, 1, 9, 5, 3, 3, 1, 1, 1, 14, 10, 6, 3, 3, 1, 1, 1, 25, 18, 12, 6, 4, 3, 1, 1, 1, 46, 29, 20, 13, 6, 4, 3, 1, 1, 1, 78, 53, 33, 20, 13, 7, 4, 3, 1, 1, 1, 136, 95, 59, 36, 22, 13, 7, 4, 3, 1, 1, 1, 242, 161, 104, 65, 36, 22, 14, 7, 4, 3, 1, 1, 1

Offset: 1

Author

Clark Kimberling, Jun 27 2004

Keywords

• nonn
• tabl

Comments

Row sums are the Carlitz sequence, A003242.

Examples

			Triangle starts
01: 1,
02: 0, 1,
03: 1, 1, 1,
04: 2, 0, 1, 1,
05: 2, 2, 1, 1, 1,
06: 5, 4, 2, 1, 1, 1,
07: 9, 5, 3, 3, 1, 1, 1,
08: 14, 10, 6, 3, 3, 1, 1, 1,
09: 25, 18, 12, 6, 4, 3, 1, 1, 1,
10: 46, 29, 20, 13, 6, 4, 3, 1, 1, 1,
11: 78, 53, 33, 20, 13, 7, 4, 3, 1, 1, 1,
12: 136, 95, 59, 36, 22, 13, 7, 4, 3, 1, 1, 1,
13: 242, 161, 104, 65, 36, 22, 14, 7, 4, 3, 1, 1, 1,
14: 419, 283, 181, 111, 67, 38, 22, 14, 7, 4, 3, 1, 1, 1,
15: 733, 500, 319, 194, 118, 68, 38, 23, 14, 7, 4, 3, 1, 1, 1,
16: 1291, 869, 557, 342, 201, 120, 70, 38, 23, 14, 7, 4, 3, 1, 1, 1,
...
T(6,1)=5 counts the compositions 1+2+1+2, 1+2+3, 1+3+2, 1+4+1, 1+5.
		

Links

• Joerg Arndt, Table of n, a(n) for n = 1..1035 (rows 1..45, flattened)

Crossrefs

Cf. A003242, A096569, A096570, A096571, A096572.

Programs

PARI

R=20;
M=matrix(R,R);
T(n,k) = if (n==0, k==0, if (k==0, n==0, M[n,k] ) );
{ for (n=1, R,
    for(k=1, n,
        M[n,k] = sum(j=0,n, T(n-k, j)) - T(n-k, k);
    );
); }
for (n=1,R,for(k=1,n, print1(M[n,k],", ") ); );
\\ Joerg Arndt, May 21 2013

Formula

Define s(0)=1, T(1, 1)=1 and T(i, j)=0 for j>i. For n>=2 and 1<=k<=n, define s(n)=T(n, 1)+T(n, 2)+...+T(n, n) and T(n, k)=s(n-k)-T(n-k, k).

G.f. for column k: C(x)*x^k/(1+x^k) where C(x) is the g.f. for A003242. - John Tyler Rascoe, May 16 2024

Extensions

Corrected by Joerg Arndt, May 21 2013