A116914 -id:A116914 - cp's OEIS frontend

A105640 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having k UUDD's, where U=(1,1) and D=(1,-1) (0<=k<=floor(n/2), n>=2). A hill in a Dyck path is a peak at level 1.

0, 1, 1, 1, 2, 3, 1, 5, 10, 3, 14, 29, 13, 1, 39, 89, 52, 6, 111, 279, 195, 36, 1, 322, 881, 722, 185, 10, 947, 2806, 2637, 867, 80, 1, 2818, 8997, 9528, 3846, 520, 15, 8470, 28997, 34163, 16382, 2976, 155, 1, 25677, 93858, 121749, 67696, 15631, 1246, 21

Offset: 2

Emeric Deutsch, May 08 2006

Row n has 1+floor(n/2) terms. Row sums are the Fine numbers (A000957). T(n,0)=A105641(n). Sum(k*T(n,k),k=0..floor(n/2))=A116914(n).

			T(5,2)=3 because we have U(UUDD)(UUDD)D, (UUDD)U(UUDD)D and U(UUDD)D(UUDD) (the UUDD's are shown between parentheses).
Triangle starts:
  0,1;
  1,1;
  2,3,1;
  5,10,3;
  14,29,13,1;
  ...

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

Cf. A000957, A105641, A116914.

G:=(1+2*z+z^2-t*z^2-sqrt(1-4*z+2*z^2-2*t*z^2+z^4-2*z^4*t+t^2*z^4))/2/z/(2+z+z^2-t*z^2)-1: Gser:=simplify(series(G,z=0,17)): for n from 2 to 14 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 2 to 14 do seq(coeff(P[n],t,j),j=0..floor(n/2)) od; # yields sequence in triangular form

G.f.: G-1, where G =G(t,z) satisfies z(2+z+z^2-tz^2)G^2-(1+2z+z^2-tz^2)G+1=0.

A126258 Triangle generated by Pascal's rule with left diagonal = [1,0,2,0,3,0,4,...] (A000027 with interpolated zeros) and right diagonal = [1,1,1,1,1,...].

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 0, 3, 2, 1, 3, 3, 5, 3, 1, 0, 6, 8, 8, 4, 1, 4, 6, 14, 16, 12, 5, 1, 0, 10, 20, 30, 28, 17, 6, 1, 5, 10, 30, 50, 58, 45, 23, 7, 1, 0, 15, 40, 80, 108, 103, 68, 30, 8, 1, 6, 15, 55, 120, 188, 211, 171, 98, 38, 9, 1, 0, 21, 70, 175, 308, 399, 382, 269, 136, 47, 10, 1

Offset: 0

Philippe Deléham, Mar 08 2007

			Triangle begins:
  1;
  0, 1;
  2, 1, 1;
  0, 3, 2, 1;
  3, 3, 5, 3, 1;
  0, 6, 8, 8, 4, 1;
  4, 6, 14, 16, 12, 5, 1;
  0, 10, 20, 30, 28, 17, 6, 1;

Cf. A034299, A116914.

T(2*n,n) = A116914(n).

Sum_{k=0..n} T(n,k) = A034299(n).

More terms from Jason Yuen, Sep 19 2024

cp's OEIS Frontend

A105640 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having k UUDD's, where U=(1,1) and D=(1,-1) (0<=k<=floor(n/2), n>=2). A hill in a Dyck path is a peak at level 1.

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