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A114596 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having abscissa of first return equal to 2k (2<=k<=n). A hill in a Dyck path is a peak at level 1. 1
1, 0, 2, 1, 0, 5, 2, 2, 0, 14, 6, 4, 5, 0, 42, 18, 12, 10, 14, 0, 132, 57, 36, 30, 28, 42, 0, 429, 186, 114, 90, 84, 84, 132, 0, 1430, 622, 372, 285, 252, 252, 264, 429, 0, 4862, 2120, 1244, 930, 798, 756, 792, 858, 1430, 0, 16796, 7338, 4240, 3110, 2604, 2394, 2376, 2574, 2860, 4862, 0, 58786 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,3

COMMENTS

Row sums are the Fine numbers (A000957). Column 2 yield the Fine numbers (A000957).

LINKS

G. C. Greubel, Rows n = 2..100 of triangle, flattened

E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.

FORMULA

T(n,n) = Catalan(n-1) (A000108).

Sum_{k=2..n} k*T(n,k) = 2*A014301(n).

T(n, k) = Catalan(k-1)*f(n-k), for 2<=k<=n, where Catalan(n) are the Catalan numbers (A000108) and f(n) = 3*Sum_{j=0..floor(n/2)} ( binomial(2n-2j, n) ) - binomial(2n+2, n+1) (the Fine numbers, A000957).

G.f.: (2*(1+x-t*x) +sqrt(1-4*x) -sqrt(1-4*t*x))/(1 +2*x +sqrt(1-4*x)) -1.

EXAMPLE

T(5,3)=2 because we have UUUDDD|UUDD and UUDUDD|UUDD, where U=(1,1), D=(1,-1) (first return is shown by a vertical bar).

Triangle begins:

   1;

   0,  2;

   1,  0,  5;

   2,  2,  0, 14;

   6,  4,  5,  0, 42;

  18, 12, 10, 14,  0, 132;

MAPLE

c:=n->binomial(2*n, n)/(n+1): f:=n->3*sum(binomial(2*n-2*j, n), j=0..floor(n/2))-binomial(2*n+2, n+1): for n from 2 to 12 do seq(c(k-1)*f(n-k), k=2..n) od; # yields sequence in triangular form

MATHEMATICA

f[n_]:= 3*Sum[Binomial[2*n-2*j, n], {j, 0, Floor[n/2]}] - Binomial[2*n+2, n +1]; Table[CatalanNumber[k-1]*f[n-k], {n, 2, 12}, {k, 2, n}] (* G. C. Greubel, Apr 06 2019 *)

PROG

(PARI) {f(n) = 3*sum(j=0, floor(n/2), binomial(2*n-2*j, n)) - binomial(2*n+2, n+1)};

for(n=2, 12, for(k=2, n, print1((binomial(2*(k-1), k)/(k-1))*f(n-k), ", "))) \\ G. C. Greubel, Apr 06 2019

(Sage)

@CachedFunction

def f(n):

  return 3*sum(binomial(2*n-2*j, n) for j in (0..floor(n/2))) - binomial(2*n+2, n+1)

def T(n, k): return catalan_number(k-1)*f(n-k)

[[T(n, k) for k in (2..n)] for n in (2..12)] # G. C. Greubel, Apr 06 2019

CROSSREFS

Cf. A000957, A000108, A014301.

Sequence in context: A133727 A103185 A130513 * A083417 A021479 A073583

Adjacent sequences:  A114593 A114594 A114595 * A114597 A114598 A114599

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Dec 12 2005

EXTENSIONS

Keyword tabf changed to tabl by Michel Marcus, Apr 09 2013

STATUS

approved

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Last modified June 1 18:43 EDT 2020. Contains 334762 sequences. (Running on oeis4.)