%I #36 Mar 11 2015 10:58:57

%S 1,0,1,0,0,1,0,0,0,1,0,0,1,0,1,0,0,0,2,0,1,0,0,0,0,3,0,1,0,0,0,1,0,4,

%T 0,1,0,0,0,0,3,0,5,0,1,0,0,0,1,0,6,0,6,0,1,0,0,0,0,3,0,10,0,7,0,1,0,0,

%U 0,0,0,7,0,15,0,8,0,1,0,0,0,0,2,0,14,0,21,0,9,0,1,0,0,0,0,0,7,0,25,0,28,0,10,0,1,0,0,0,0,1,0,17,0,41,0,36,0,11,0,1

%N Triangle read by rows: T(n,k) is the number of Dyck paths of semilength k such that the area between the x-axis and the path is n (n>=0; 0<=k<=n).

%C Triangle A129182 transposed.

%C Column sums give the Catalan numbers (A000108).

%C Row sums give A143951.

%C Sums along falling diagonals give A005169.

%C T(4n,2n) = A240008(n). - _Alois P. Heinz_, Mar 30 2014

%H Joerg Arndt and Alois P. Heinz, <a href="/A239927/b239927.txt">Rows n = 0..140, flattened</a>

%F G.f.: F(x,y) satisfies F(x,y) = 1 / (1 - x*y * F(x, x^2*y) ).

%F G.f.: 1/(1 - y*x/(1 - y*x^3/(1 - y*x^5/(1 - y*x^7/(1 - y*x^9/( ... )))))).

%e Triangle begins:

%e 00: 1;

%e 01: 0, 1;

%e 02: 0, 0, 1;

%e 03: 0, 0, 0, 1;

%e 04: 0, 0, 1, 0, 1;

%e 05: 0, 0, 0, 2, 0, 1;

%e 06: 0, 0, 0, 0, 3, 0, 1;

%e 07: 0, 0, 0, 1, 0, 4, 0, 1;

%e 08: 0, 0, 0, 0, 3, 0, 5, 0, 1;

%e 09: 0, 0, 0, 1, 0, 6, 0, 6, 0, 1;

%e 10: 0, 0, 0, 0, 3, 0, 10, 0, 7, 0, 1;

%e 11: 0, 0, 0, 0, 0, 7, 0, 15, 0, 8, 0, 1;

%e 12: 0, 0, 0, 0, 2, 0, 14, 0, 21, 0, 9, 0, 1;

%e 13: 0, 0, 0, 0, 0, 7, 0, 25, 0, 28, 0, 10, 0, 1;

%e 14: 0, 0, 0, 0, 1, 0, 17, 0, 41, 0, 36, 0, 11, 0, 1;

%e 15: 0, 0, 0, 0, 0, 5, 0, 35, 0, 63, 0, 45, 0, 12, 0, 1;

%e 16: 0, 0, 0, 0, 1, 0, 16, 0, 65, 0, 92, 0, 55, 0, 13, 0, 1;

%e 17: 0, 0, 0, 0, 0, 5, 0, 40, 0, 112, 0, 129, 0, 66, 0, 14, 0, 1;

%e 18: 0, 0, 0, 0, 0, 0, 16, 0, 86, 0, 182, 0, 175, 0, 78, 0, 15, 0, 1;

%e 19: 0, 0, 0, 0, 0, 3, 0, 43, 0, 167, 0, 282, 0, 231, 0, 91, 0, 16, 0, 1;

%e 20: 0, 0, 0, 0, 0, 0, 14, 0, 102, 0, 301, 0, 420, 0, 298, 0, 105, 0, 17, 0, 1;

%e ...

%e Column k=4 corresponds to the following 14 paths (dots denote zeros):

%e #: path area steps (Dyck word)

%e 01: [ . 1 . 1 . 1 . 1 . ] 4 + - + - + - + -

%e 02: [ . 1 . 1 . 1 2 1 . ] 6 + - + - + + - -

%e 03: [ . 1 . 1 2 1 . 1 . ] 6 + - + + - - + -

%e 04: [ . 1 . 1 2 1 2 1 . ] 8 + - + + - + - -

%e 05: [ . 1 . 1 2 3 2 1 . ] 10 + - + + + - - -

%e 06: [ . 1 2 1 . 1 . 1 . ] 6 + + - - + - + -

%e 07: [ . 1 2 1 . 1 2 1 . ] 8 + + - - + + - -

%e 08: [ . 1 2 1 2 1 . 1 . ] 8 + + - + - - + -

%e 09: [ . 1 2 1 2 1 2 1 . ] 10 + + - + - + - -

%e 10: [ . 1 2 1 2 3 2 1 . ] 12 + + - + + - - -

%e 11: [ . 1 2 3 2 1 . 1 . ] 10 + + + - - - + -

%e 12: [ . 1 2 3 2 1 2 1 . ] 12 + + + - - + - -

%e 13: [ . 1 2 3 2 3 2 1 . ] 14 + + + - + - - -

%e 14: [ . 1 2 3 4 3 2 1 . ] 16 + + + + - - - -

%e There are no paths with weight < 4, one with weight 4, none with weight 5, 3 with weight 6, etc., therefore column k=4 is

%e [0, 0, 0, 0, 1, 0, 3, 0, 3, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 0, ...].

%e Row n=8 is [0, 0, 0, 0, 3, 0, 5, 0, 1], the corresponding paths of weight=8 are:

%e Semilength 4:

%e [ . 1 . 1 2 1 2 1 . ]

%e [ . 1 2 1 . 1 2 1 . ]

%e [ . 1 2 1 2 1 . 1 . ]

%e Semilength 6:

%e [ . 1 . 1 . 1 . 1 . 1 2 1 . ]

%e [ . 1 . 1 . 1 . 1 2 1 . 1 . ]

%e [ . 1 . 1 . 1 2 1 . 1 . 1 . ]

%e [ . 1 . 1 2 1 . 1 . 1 . 1 . ]

%e [ . 1 2 1 . 1 . 1 . 1 . 1 . ]

%e Semilength 8:

%e [ . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . ]

%p b:= proc(x, y, k) option remember;

%p `if`(y<0 or y>x or k<0, 0, `if`(x=0, `if`(k=0, 1, 0),

%p b(x-1, y-1, k-y+1/2)+ b(x-1, y+1, k-y-1/2)))

%p end:

%p T:= (n, k)-> b(2*k, 0, n):

%p seq(seq(T(n, k), k=0..n), n=0..20); # _Alois P. Heinz_, Mar 29 2014

%t b[x_, y_, k_] := b[x, y, k] = If[y<0 || y>x || k<0, 0, If[x == 0, If[k == 0, 1, 0], b[x-1, y-1, k-y+1/2] + b[x-1, y+1, k-y-1/2]]]; T[n_, k_] := b[2*k, 0, n]; Table[ Table[T[n, k], {k, 0, n}], {n, 0, 20}] // Flatten (* _Jean-François Alcover_, Feb 18 2015, after _Alois P. Heinz_ *)

%o (PARI)

%o rvec(V) = { V=Vec(V); my(n=#V); vector(n, j, V[n+1-j] ); }

%o print_triangle(V)= { my( N=#V ); for(n=1, N, print( rvec( V[n]) ) ); }

%o N=20; x='x+O('x^N);

%o F(x,y, d=0)=if (d>N, 1, 1 / (1-x*y * F(x, x^2*y, d+1) ) );

%o v= Vec( F(x,y) );

%o print_triangle(v)

%Y Sequences obtained by particular choices for x and y in the g.f. F(x,y) are: A000108 (F(1, x)), A143951 (F(x, 1)), A005169 (F(sqrt(x), sqrt(x))), A227310 (1+x*F(x, x^2), also 2-1/F(x, 1)), A239928 (F(x^2, x)), A052709 (x*F(1,x+x^2)), A125305 (F(1, x+x^3)), A002212 (F(1, x/(1-x))).

%Y Cf. A047998, A138158, A227543.

%Y Cf. A129181.

%K nonn,tabl

%O 0,19

%A _Joerg Arndt_, Mar 29 2014