The On-Line Encyclopedia of Integer Sequences (OEIS)

A233323 revision #25

A233323

Triangle read by rows: T(n,k) = number of palindromic compositions of n in which the largest part is equal to k, 1 <= k <= n.

1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 1, 0, 1, 1, 4, 1, 1, 0, 1, 1, 2, 3, 0, 1, 0, 1, 1, 7, 3, 3, 0, 1, 0, 1, 1, 4, 6, 1, 2, 0, 1, 0, 1, 1, 12, 7, 7, 1, 2, 0, 1, 0, 1, 1, 7, 12, 3, 5, 0, 2, 0, 1, 0, 1, 1, 20, 16, 15, 3, 5, 0, 2, 0, 1, 0, 1

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OFFSET

1,8

COMMENTS

A palindromic composition of a natural number m is an ordered partition of m into N+1 natural numbers (or parts), p_0, p_1, ..., p_N, of the form m = p_0 + p_1 + ... + p_N such that p_j = p_{N-j}, for each j in {0,...,N}. Two palindromic compositions, sum_{j=0..N} p_j and sum_{j=0..N} q_j (say), are identical if and only if p_j = q_j, j = 0,...,N; otherwise they are taken to be distinct.

LINKS

Table of n, a(n) for n=1..78.

Charles R Greathouse IV and Alois P. Heinz, Rows n = 1..141, flattened (rows n = 1..50 from Charles R Greathouse IV)

V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

EXAMPLE

There are eight palindromic compositions of n=7, namely, {7}, {3,1,3}, {2,3,2}, {2,1,1,1,2}, {1,5,1}, {1,2,1,2,1}, {1,1,3,1,1}, {1,1,1,1,1,1,1}, and three of them have the largest part equal to 3, so T(7,3) = 3.

Triangle T(n,k) begins:

1, 1;

1, 0, 1,

1, 2, 0, 1;

1, 1, 1, 0, 1;

1, 4, 1, 1, 0, 1;

1, 2, 3, 0, 1, 0, 1;

1, 7, 3, 3, 0, 1, 0, 1;

1, 4, 6, 1, 2, 0, 1, 0, 1;

1, 12, 7, 7, 1, 2, 0, 1, 0, 1;

MAPLE

b:= proc(n, k) option remember; `if`(n<=k, 1, 0)+

add(b(n-2*j, k), j=1..min(k, iquo(n, 2)))

end:

T:= (n, k)-> b(n, k) -b(n, k-1):

seq(seq(T(n, k), k=1..n), n=1..14); # Alois P. Heinz, Dec 11 2013

MATHEMATICA

b[n_, k_] := b[n, k] = If[n <= k, 1, 0] + Sum[b[n-2*j, k], { j, 1, Min[k, Quotient[n, 2]]}]; t[n_, k_] := b[n, k] - b[n, k-1]; Table[Table[t[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Alois P. Heinz's Maple code *)

PROG

(PARI) T(n, k, ok=0)={

if(n<1, return(n==0 && ok));

if(k>n/2 && !ok,

n-=k;

if(n<0||n%2, return(0));

return(2^max(n/2-1, 0))

);

sum(i=1, k,

T(n-2*i, k, ok||i==k)

)+(n==k || (ok && n<k))

}; \\ Charles R Greathouse IV, Dec 11 2013