OFFSET
1,3
COMMENTS
By changing the upper summation index in the recurrence from k-1 to n-1 we get the Motzkin numbers A001006.
That is, by changing
Sum_{i=1..k-1} t(n-i, k-1) - Sum_{i=1..k-1} t(n-i, k)
into
Sum_{i=1..n-1} t(n-i, k-1) - Sum_{i=1..n-1} t(n-i, k),
we get the Motzkin numbers.
MATHEMATICA
Clear[t, n, k, i, nn, x];
coeff = {1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0};
mp[m_, e_] :=
If[e == 0, IdentityMatrix@Length@m, MatrixPower[m, e]]; nn =
Length[coeff]; cc = Range[nn]*0 + 1; Monitor[
Do[Clear[t]; t[n_, 1] := t[n, 1] = cc[[n]];
t[n_, k_] :=
t[n, k] =
If[n >= k,
Sum[t[n - i, k - 1], {i, 1, k - 1}] -
Sum[t[n - i, k], {i, 1, k - 1}], 0];
A4 = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}];
A5 = A4[[1 ;; nn - 1]]; A5 = Prepend[A5, ConstantArray[0, nn]];
cc = Total[
Table[coeff[[n]]*mp[A5, n - 1][[All, 1]], {n, 1, nn}]]; , {i, 1,
nn}], i]; cc
CROSSREFS
KEYWORD
nonn
AUTHOR
Mats Granvik, Dec 30 2017
STATUS
approved