A212363 - OEIS

%I #20 Jan 21 2019 17:07:20

%S 1,1,1,1,1,1,1,1,2,1,1,1,1,5,1,1,1,1,2,14,1,1,1,1,1,4,42,1,1,1,1,1,2,

%T 8,132,1,1,1,1,1,1,4,17,429,1,1,1,1,1,1,2,7,37,1430,1,1,1,1,1,1,1,4,

%U 12,82,4862,1,1,1,1,1,1,1,2,7,22,185,16796,1

%N Number A(n,k) of Dyck n-paths all of whose ascents and descents have lengths equal to 1+k*m (m>=0); square array A(n,k), n>=0, k>=0, read by antidiagonals.

%H Alois P. Heinz, <a href="/A212363/b212363.txt">Antidiagonals n = 0..140, flattened</a>

%F G.f. of column k>0 satisfies: A_k(x) = 1+A_k(x)*(x-x^k*(1-A_k(x))), g.f. of column k=0: A_0(x) = 1/(1-x).

%F A(n,k) = A(n-1,k) + Sum_{j=1..n-k} A(j,k)*A(n-k-j,k) for n,k>0; A(n,0) = A(0,k) = 1.

%F G.f. of column k > 0: (1 - x + x^k - sqrt((1 - x + x^k)^2 - 4*x^k)) / (2*x^k). - _Vaclav Kotesovec_, Sep 02 2014

%e A(3,0) = 1: UDUDUD.

%e A(3,1) = 5: UDUDUD, UDUUDD, UUDDUD, UUDUDD, UUUDDD.

%e A(4,2) = 4: UDUDUDUD, UDUUUDDD, UUUDDDUD, UUUDUDDD.

%e A(5,2) = 8: UDUDUDUDUD, UDUDUUUDDD, UDUUUDDDUD, UDUUUDUDDD, UUUDDDUDUD, UUUDUDDDUD, UUUDUDUDDD, UUUUUDDDDD.

%e A(5,3) = 4: UDUDUDUDUD, UDUUUUDDDD, UUUUDDDDUD, UUUUDUDDDD.

%e Square array A(n,k) begins:

%e   1,   1,  1,  1,  1,  1,  1,  1, ...

%e   1,   1,  1,  1,  1,  1,  1,  1, ...

%e   1,   2,  1,  1,  1,  1,  1,  1, ...

%e   1,   5,  2,  1,  1,  1,  1,  1, ...

%e   1,  14,  4,  2,  1,  1,  1,  1, ...

%e   1,  42,  8,  4,  2,  1,  1,  1, ...

%e   1, 132, 17,  7,  4,  2,  1,  1, ...

%e   1, 429, 37, 12,  7,  4,  2,  1, ...

%p A:= proc(n, k) option remember;

%p       `if`(k=0, 1, `if`(n=0, 1, A(n-1, k)

%p                    +add(A(j, k)*A(n-k-j, k), j=1..n-k)))

%p     end:

%p seq(seq(A(n, d-n), n=0..d), d=0..15);

%p # second Maple program:

%p A:= (n, k)-> `if`(k=0, 1, coeff(series(RootOf(

%p               A||k=1+A||k*(x-x^k*(1-A||k)), A||k), x, n+1), x, n)):

%p seq(seq(A(n, d-n), n=0..d), d=0..15);

%t A[n_, k_] := A[n, k] = If[k == 0, 1, If[n == 0, 1, A[n-1, k] + Sum[A[j, k]*A[n-k-j, k], {j, 1, n-k}]]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 15}] // Flatten (* _Jean-François Alcover_, Jan 15 2014, translated from first Maple program *)

%Y Columns k=0-10 give: A000012, A000108, A004148, A023432, A023427, A212364, A212365, A212366, A212367, A212368, A212369.

%K nonn,tabl

%O 0,9

%A _Alois P. Heinz_, May 10 2012