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A114462 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k ascents of length 2 starting at an even level (0<=k<=floor(n/2)). 5
1, 1, 1, 1, 2, 3, 6, 7, 1, 18, 19, 5, 54, 59, 18, 1, 166, 191, 65, 7, 522, 631, 242, 34, 1, 1670, 2123, 906, 154, 9, 5418, 7247, 3395, 680, 55, 1, 17786, 25011, 12746, 2932, 300, 11, 58974, 87071, 47931, 12414, 1540, 81, 1, 197226, 305275, 180439, 51878, 7552 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row n has 1+floor(n/2) terms. Row sums are the Catalan numbers (A000108). Sum(kT(n,k), k=0..floor(n/2)) = binomial(2n-3,n-1)-binomial(2n-4,n) = A077587(n-2) (n>=2). Column 0 yields A114464.

LINKS

Alois P. Heinz, Rows n = 0..200, flattened

FORMULA

G.f.: G(t,z) satisfies zG^2-(1-z+tz-3tz^2+3z^2-z^3-t^2z^3+2tz^3)G+1-z+z^2+tz-tz^2=0.

EXAMPLE

T(4,1) = 7 because we have (UU)DDUDUD, UD(UU)DDUD, UDUD(UU)DD, (UU)DUDDUD,

UD(UU)DUDD, (UU)DUDUDD and (UU)DUUDDD, where U=(1,1), D=(1,-1) (the ascents of length 2 starting at an even level are shown between parentheses; note that the last path has an ascent of length 2 that starts at an odd level).

Triangle starts:

1;

1;

1,   1;

2,   3;

6,   7,  1;

18, 19,  5;

54, 59, 18, 1;

MAPLE

G:= 1/2/z*(3*z^2+2*z^3*t+1-z^3*t^2-3*z^2*t-z^3+t*z-z -sqrt(1+20*z^3*t-18*z^5*t^2+15*z^4*t^2+18*z^5*t+6*z^5*t^3-2*z^4*t^3-12*z^2*t -12*z^3 -6*z-24*z^4*t-8*z^3*t^2+z^6-6*z^5+11*z^4 +z^2*t^2+6*z^6*t^2 -4*z^6*t^3 -4*z^6*t+z^6*t^4+2*t*z +11*z^2)): Gser:=simplify(series(G, z=0, 17)): P[0]:=1: for n from 1 to 14 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 14 do seq(coeff(t*P[n], t^j), j=1..1+floor(n/2)) od; # yields sequence in triangular form

# second Maple program:

b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0, `if`(x=0,

      `if`(t=2, z, 1), expand(b(x-1, y-1, min(3, t+1))+

      `if`(t=2 and irem(y, 2)=0, z, 1)*b(x-1, y+1, 0))))

    end:

T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0$2)):

seq(T(n), n=0..14);  # Alois P. Heinz, Mar 12 2014

MATHEMATICA

b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x==0, If[t==2, z, 1], Expand[ b[x-1, y-1, Min[3, t+1]] + If[t==2 && Mod[y, 2]==0, z, 1]*b[x-1, y+1, 0]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-Fran├žois Alcover, Mar 31 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A077587, A000108, A114463, A114464, A114465, A102402.

Sequence in context: A182560 A298750 A001058 * A169746 A234613 A258996

Adjacent sequences:  A114459 A114460 A114461 * A114463 A114464 A114465

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Nov 29 2005

STATUS

approved

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Last modified February 21 22:56 EST 2018. Contains 299427 sequences. (Running on oeis4.)