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

1,3

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.

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

Links

Table of n, a(n) for n=1..30.

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

Cf. A001006, A239605.

Sequence in context: A130616 A089324 A339159 * A111759 A305751 A047749

Adjacent sequences: A297435 A297436 A297437 * A297439 A297440 A297441

Keyword

nonn

Author

Mats Granvik, Dec 30 2017

Status

approved