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A189912 Extended Motzkin numbers, Sum_{k>=0} C(n,k)C(k), C(k) the extended Catalan number A057977(k). 8
1, 2, 4, 10, 25, 66, 177, 484, 1339, 3742, 10538, 29866, 85087, 243478, 699324, 2015082, 5822619, 16865718, 48958404, 142390542, 414837699, 1210439958, 3536809521, 10347314544, 30306977757, 88861597426, 260798283502, 766092871654, 2252240916665 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) = Sum_{k=0..n} binomial(n,k)*A057977(k). For comparison:

A001006(n) = Sum_{k=0..n} binomial(n,k)*A057977(k)*[k is even],

A005717(n) = Sum_{k=0..n} binomial(n,k)*A057977(k)*[k is odd].

Thus one might simply say: The extended Motzkin numbers are the binomial sum of the extended Catalan numbers. Moreover: The Catalan numbers aerated with 0's at odd positions (A126120) are the inverse binomial transform of the Motzkin numbers (A001006). The complementary Catalan numbers (A001700) aerated with 0's at even positions (A138364) are the inverse binomial transform of the complementary Motzkin numbers (A005717). The extended Catalan numbers (A057977 = A126120 + A138364) are the inverse binomial transform of the extended Motzkin numbers (A189912).

David Scambler observed that [1, a(n-1)] for n >= 1 count the Dyck paths of semilength n which satisfy the condition "number of peaks <= number of returns + number of hills". - Peter Luschny, Oct 22 2012

REFERENCES

Peter Luschny, Divide, swing and conquer the factorial and the lcm{1,2,...,n}, preprint, April 2008.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

A. Asinowski, G. Rote, Point sets with many non-crossing matchings, arXiv preprint arXiv:1502.04925 [cs.CG], 2015.

Peter Luschny, The Scambler_statistic_on_Dyck_words.

FORMULA

a(n) = Sum_{k=0..n}(n!/(((n-k)!*floor(k/2)!^2)*(floor(k/2)+1)).

Recurrence: (n+2)*(n^2 + 2*n - 5)*a(n) = (2*n^3 + 7*n^2 - 14*n - 7)*a(n-1) + 3*(n-1)*(n^2 + 4*n - 2)*a(n-2). - Vaclav Kotesovec, Mar 20 2014

a(n) ~ 3^(n+1/2) / (2*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014

Conjecture: a(n) = Sum_{k=0..floor(n/2)} (n+1-2*k)*A055151(n,k). - Werner Schulte, Oct 23 2016

MAPLE

A189912 := proc(n) local k;

add(n!/(((n-k)!*iquo(k, 2)!^2)*(iquo(k, 2)+1)), k=0..n) end:

M := proc(n) option remember; `if`(n<2, 1, (3*(n-1)*M(n-2)+(2*n+1)*M(n-1))/(n+2)) end:

A189912 := n -> n*M(n-1)+M(n);

seq(A189912(i), i=0..28); # Peter Luschny, Sep 12 2011

MATHEMATICA

A057977[n_] := n!/(Quotient[n, 2]!^2*(Quotient[n, 2] + 1)); a[n_] := Sum[Binomial[n, k]*A057977[k], {k, 0, n}]; Table[a[n], {n, 0, 28}] (* Jean-Fran├žois Alcover, May 21 2013, after Peter Luschny *)

Table[Sum[n!/(((n - k)!*Floor[k/2]!^2)*(Floor[k/2] + 1)), {k, 0, n}], {n, 0, 100}] (* G. C. Greubel, Jan 24 2017 *)

PROG

(Sage)

@CachedFunction

def M(n): return (3*(n-1)*M(n-2)+(2*n+1)*M(n-1))/(n+2) if n>1 else 1

A189912 = lambda n: n*M(n-1) + M(n)

[A189912(i) for i in (0..28)] # Peter Luschny, Oct 22 2012

(PARI) for(n=0, 50, print1(sum(k=0, n, n!/((n-k)!*((floor(k/2))!)^2*(floor(k/2) + 1))), ", ")) \\ G. C. Greubel, Jan 24 2017

CROSSREFS

Cf. A001700, A005717, A001006, A057977, A126120, A138364, A217539, A217540.

Sequence in context: A166516 A230552 A230555 * A268321 A195981 A124500

Adjacent sequences:  A189909 A189910 A189911 * A189913 A189914 A189915

KEYWORD

nonn

AUTHOR

Peter Luschny, May 01 2011

STATUS

approved

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Last modified February 24 06:13 EST 2018. Contains 299597 sequences. (Running on oeis4.)