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

1,7

Comments

Row sums are the Carlitz sequence, A003242.

Links

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

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

Example

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.

Prog

(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

Crossrefs

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

Sequence in context: A348691 A290885 A059881 * A025862 A004540 A353945

Adjacent sequences: A096565 A096566 A096567 * A096569 A096570 A096571

Keyword

nonn,tabl

Author

Clark Kimberling, Jun 27 2004

Extensions

Corrected by Joerg Arndt, May 21 2013

Status

approved