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A297438 A divisor analog of the Motzkin numbers A001006. 0
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 4 01:04 EDT 2022. Contains 357237 sequences. (Running on oeis4.)