A297438 A divisor analog of the Motzkin numbers A001006.

1, 1, 2, 3, 7, 12, 29, 56, 134, 283, 672, 1496, 3568, 8214, 19678, 46364, 111766, 267467, 648941, 1570540, 3833777, 9357181, 22967808, 56430230, 139193762, 343825265, 851777363, 2113382992, 5255584309, 13089273904

Offset: 1

Author

Mats Granvik, Dec 30 2017

Keywords

• nonn

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.

With this change of upper summation index, a(n) is to A001006 as A239605 is to A000108.

Crossrefs

Cf. A001006, A239605.

Programs

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