A309938 Triangle read by rows: T(n,k) is the number of compositions of n with k parts and differences all equal to 1 or -1.

1, 1, 0, 1, 2, 0, 1, 0, 1, 0, 1, 2, 1, 0, 0, 1, 0, 2, 2, 0, 0, 1, 2, 1, 0, 1, 0, 0, 1, 0, 1, 4, 1, 0, 0, 0, 1, 2, 2, 0, 3, 2, 0, 0, 0, 1, 0, 1, 4, 2, 0, 1, 0, 0, 0, 1, 2, 1, 0, 3, 6, 1, 0, 0, 0, 0, 1, 0, 2, 4, 3, 0, 4, 2, 0, 0, 0, 0, 1, 2, 1, 0, 3, 8, 3, 0, 1, 0, 0, 0, 0

Offset

1,5

Comments

Parts will alternate between being odd and even. For even k, a composition cannot be the same as its reversal and therefore for even k, T(n,k) is even.

Links

Alois P. Heinz, Rows n = 1..200, flattened

Example

Triangle begins:
  1;
  1, 0;
  1, 2, 0;
  1, 0, 1, 0;
  1, 2, 1, 0, 0;
  1, 0, 2, 2, 0,  0;
  1, 2, 1, 0, 1,  0, 0;
  1, 0, 1, 4, 1,  0, 0, 0;
  1, 2, 2, 0, 3,  2, 0, 0, 0;
  1, 0, 1, 4, 2,  0, 1, 0, 0, 0;
  1, 2, 1, 0, 3,  6, 1, 0, 0, 0, 0;
  1, 0, 2, 4, 3,  0, 4, 2, 0, 0, 0, 0;
  1, 2, 1, 0, 3,  8, 3, 0, 1, 0, 0, 0, 0;
  1, 0, 1, 4, 3,  0, 6, 8, 1, 0, 0, 0, 0, 0;
  1, 2, 2, 0, 4, 10, 5, 0, 5, 2, 0, 0, 0, 0, 0;
  ...
For n = 6 there are a total of 5 compositions:
  k = 1: (6)
  k = 3: (123), (321)
  k = 4: (2121), (1212)

Maple

b:= proc(n, i) option remember; `if`(n<1 or i<1, 0,
     `if`(n=i, x, add(expand(x*b(n-i, i+j)), j=[-1, 1])))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(add(b(n, j), j=1..n)):
seq(T(n), n=1..14);  # Alois P. Heinz, Jul 22 2023

Mathematica

b[n_, i_] := b[n, i] = If[n < 1 || i < 1, 0, If[n == i, x, Sum[Expand[x*b[n - i, i + j]], {j, {-1, 1}}]]];
T[n_] :=  CoefficientList[Sum[b[n, j], {j, 1, n}], x] // Rest // PadRight[#, n]&;
Table[T[n], {n, 1, 13}] // Flatten (* Jean-François Alcover, Sep 06 2023, after Alois P. Heinz *)

Prog

(PARI)
step(R, n)={matrix(n, n, i, j, if(i>j, if(j>1, R[i-j, j-1]) + if(j+1<=n, R[i-j, j+1])) )}
T(n)={my(v=vector(n), R=matid(n), m=0); while(R, m++; v[m]+=vecsum(R[n, ]); R=step(R, n)); v}
for(n=1, 15, print(T(n)))

Crossrefs

Row sums are A173258.

T(2n,n) gives A364529.

Cf. A309931, A309937, A309939, A325557.

Sequence in context: A116927 A137276 A287234 * A140581 A137277 A039975

Adjacent sequences: A309935 A309936 A309937 * A309939 A309940 A309941

Keyword

nonn,tabl

Author

Andrew Howroyd, Aug 23 2019

Status

approved