login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A191521 Triangle read by rows: T(n,k) is the number of left factors of Dyck paths of length n that have k valleys (i.e., a (1,-1)-step followed by a (1,1)-step). 3
1, 1, 2, 2, 1, 3, 3, 3, 6, 1, 4, 12, 4, 4, 18, 12, 1, 5, 30, 30, 5, 5, 40, 60, 20, 1, 6, 60, 120, 60, 6, 6, 75, 200, 150, 30, 1, 7, 105, 350, 350, 105, 7, 7, 126, 525, 700, 315, 42, 1, 8, 168, 840, 1400, 840, 168, 8, 8, 196, 1176, 2450, 1960, 588, 56, 1, 9, 252, 1764, 4410, 4410, 1764, 252, 9 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Row n>=1 contains ceiling(n/2) entries.

Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).

Sum_{k>=0} k*T(n,k) = A191522(n).

LINKS

Alois P. Heinz, Rows n = 0..300, flattened

FORMULA

G.f.: G(t,z) = (1+t*z^2-z^2-Q)/(t*z*(t*z^2-1+2*z-z^2+Q)), where Q = sqrt(((1-z)^2-t*z^2)*((1+z)^2-t*z^2)).

T(n,k) = 2*C(n/2,k)*C(n/2,k+1)*(n/2+1)/n, for even n, C((n+1)/2,k+1)*Sum_{j=1..(n+1)/2} (-1)^(j-1)*C((n+1)/2,k-j+1), for odd n, T(0,0)=1. - Vladimir Kruchinin, Jul 24 2019

EXAMPLE

T(4,1)=3 because we have U(DU)D, U(DU)U, and UU(DU), where U=(1,1) and D=(1,-1) (the valleys are shown between parentheses).

Triangle starts:

  1;

  1;

  2;

  2,  1;

  3,  3;

  3,  6,  1;

  4, 12,  4;

  4, 18, 12,  1;

  ...

MAPLE

Q := sqrt(((1-z)^2-t*z^2)*((1+z)^2-t*z^2)): G := (1+t*z^2-z^2-Q)/(t*z*(t*z^2-1+2*z-z^2+Q)): Gser := simplify(series(G, z = 0, 19)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: 1; for n to 16 do seq(coeff(P[n], t, k), k = 0 .. ceil((1/2)*n)-1) end do; # yields sequence in triangular form

# second Maple program:

b:= proc(x, y, t) option remember; expand(`if`(x=0, 1,

     `if`(y>0, b(x-1, y-1, z), 0)+b(x-1, y+1, 1)*t))

    end:

T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0, 1)):

seq(T(n), n=0..30);  # Alois P. Heinz, Mar 29 2017

MATHEMATICA

T[n_, m_] := If [n == 0 && m == 0, 1, If[n == 0, 0, If[OddQ[n-1], (2* Binomial[n/2, m]*Binomial[n/2, m+1]*(n/2 + 1))/n, Binomial[(n+1)/2, m+1]*Sum[(-1)^(k-1)*Binomial[(n+1)/2, m-k+1], {k, 1, (n+1)/2}]]]];

Table[T[n, m], {n, 0, 16}, {m, 0, If[n <= 2, 0, Quotient[n-1, 2]]}] // Flatten (* Jean-François Alcover, Feb 16 2021, after Vladimir Kruchinin *)

PROG

(Maxima)

T(n, m):=if n=0 and m=0 then 1 else if n=0 then 0 else if oddp(n-1) then (2*binomial(n/2, m)*binomial(n/2, m+1)*(n/2+1))/n else binomial((n+1)/2, m+1)*sum((-1)^(k-1)*binomial((n+1)/2, m-k+1), k, 1, (n+1)/2);

/* Vladimir Kruchinin, Jul 24 2019 */

CROSSREFS

Cf. A001405, A124428, A191522.

Sequence in context: A112209 A240127 A109524 * A245370 A321341 A284549

Adjacent sequences:  A191518 A191519 A191520 * A191522 A191523 A191524

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Jun 05 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 6 22:42 EST 2021. Contains 349567 sequences. (Running on oeis4.)