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A212363 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.

%I A212363 #20 Jan 21 2019 17:07:20
%S A212363 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 A212363 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 A212363 12,82,4862,1,1,1,1,1,1,1,2,7,22,185,16796,1
%N A212363 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 A212363 Alois P. Heinz, <a href="/A212363/b212363.txt">Antidiagonals n = 0..140, flattened</a>
%F A212363 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 A212363 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 A212363 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 A212363 A(3,0) = 1: UDUDUD.
%e A212363 A(3,1) = 5: UDUDUD, UDUUDD, UUDDUD, UUDUDD, UUUDDD.
%e A212363 A(4,2) = 4: UDUDUDUD, UDUUUDDD, UUUDDDUD, UUUDUDDD.
%e A212363 A(5,2) = 8: UDUDUDUDUD, UDUDUUUDDD, UDUUUDDDUD, UDUUUDUDDD, UUUDDDUDUD, UUUDUDDDUD, UUUDUDUDDD, UUUUUDDDDD.
%e A212363 A(5,3) = 4: UDUDUDUDUD, UDUUUUDDDD, UUUUDDDDUD, UUUUDUDDDD.
%e A212363 Square array A(n,k) begins:
%e A212363   1,   1,  1,  1,  1,  1,  1,  1, ...
%e A212363   1,   1,  1,  1,  1,  1,  1,  1, ...
%e A212363   1,   2,  1,  1,  1,  1,  1,  1, ...
%e A212363   1,   5,  2,  1,  1,  1,  1,  1, ...
%e A212363   1,  14,  4,  2,  1,  1,  1,  1, ...
%e A212363   1,  42,  8,  4,  2,  1,  1,  1, ...
%e A212363   1, 132, 17,  7,  4,  2,  1,  1, ...
%e A212363   1, 429, 37, 12,  7,  4,  2,  1, ...
%p A212363 A:= proc(n, k) option remember;
%p A212363       `if`(k=0, 1, `if`(n=0, 1, A(n-1, k)
%p A212363                    +add(A(j, k)*A(n-k-j, k), j=1..n-k)))
%p A212363     end:
%p A212363 seq(seq(A(n, d-n), n=0..d), d=0..15);
%p A212363 # second Maple program:
%p A212363 A:= (n, k)-> `if`(k=0, 1, coeff(series(RootOf(
%p A212363               A||k=1+A||k*(x-x^k*(1-A||k)), A||k), x, n+1), x, n)):
%p A212363 seq(seq(A(n, d-n), n=0..d), d=0..15);
%t A212363 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 A212363 Columns k=0-10 give: A000012, A000108, A004148, A023432, A023427, A212364, A212365, A212366, A212367, A212368, A212369.
%K A212363 nonn,tabl
%O A212363 0,9
%A A212363 _Alois P. Heinz_, May 10 2012