A214247 Number A(n,k) of compositions of n where differences between neighboring parts are in {-k,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 4, 4, 1, 1, 1, 1, 1, 2, 5, 2, 1, 1, 1, 1, 1, 3, 3, 5, 4, 1, 1, 1, 1, 1, 1, 2, 2, 7, 3, 1, 1, 1, 1, 1, 1, 3, 3, 6, 10, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 9, 2

Offset

0,6

Links

Alois P. Heinz, Antidiagonals n = 0..140

Formula

G.f. for column k > 0: A(k,q) is A(k,q,t) = Sum_{n,m>=0} (q^n)*(t^m) under the transform (q^n)*(t^m) -> (q^n)/(1-q^m) for all m > 0 where A(k,q,t) = 1 + Sum_{i>=0} ( t^((2*i)+1) * Cw(i,k,q) * (Sum_{j>=0} (Product_{u=1..j} (Sum_{v>=0} t^((2*v)+1) * q^(((2*v)+1)*k*u) * Cw(v,k,q))))^2 ), Cw(i,k,q) = q^(((k+2)*i)+1) * Ca(i,q^(2*k)), and Ca(i,q) is the i-th Carlitz-Riordan q-Catalan number (i-th row polynomial of A227543). - John Tyler Rascoe, Sep 13 2024

Example

A(5,0) = 2: [5], [1,1,1,1,1].
A(5,1) = 4: [5], [3,2], [2,3], [2,1,2].
A(5,2) = 2: [5], [1,3,1].
A(5,3) = 3: [5], [4,1], [1,4].
Square array A(n,k) begins:
  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
  2,  1,  1,  1,  1,  1,  1,  1,  1, ...
  2,  3,  1,  1,  1,  1,  1,  1,  1, ...
  3,  2,  3,  1,  1,  1,  1,  1,  1, ...
  2,  4,  2,  3,  1,  1,  1,  1,  1, ...
  4,  5,  3,  2,  3,  1,  1,  1,  1, ...
  2,  5,  2,  3,  2,  3,  1,  1,  1, ...
  4,  7,  6,  1,  3,  2,  3,  1,  1, ...

Maple

b:= proc(n, i, k) option remember;
      `if`(n<1 or i<1, 0, `if`(n=i, 1, add(b(n-i, i+j, k), j={-k, k})))
    end:
A:= (n, k)-> `if`(n=0, 1, add(b(n, j, k), j=1..n)):
seq(seq(A(n, d-n), n=0..d), d=0..15);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n < 1 || i < 1, 0, If[n == i, 1, Sum[b[n - i, i + j, k], { j, Union[{-k, k}]}]]]; a[n_, k_] := If[n == 0, 1, Sum[b[n, j, k], {j, 1, n}]]; Table[Table[a[n, d - n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

Prog

(PARI)
tri(k) = {(k*(k+1)/2)}
ra(n) = {(sqrt(1+8*n)-1)/2}
C(q, n) = {c = if(n<=1, return(1)); return(sum(i=0, n-1, C(q, i) * C(q, n-1-i) * q^((i+1)*(n-1 -i)))); }
Cw_q(i, k) = {return(q^(((k+2)*i)+1) * polrecip(C(q^(2*k), i)))}
A_qt(k, N) = {1 + sum(i=0, N/(k+1), t^((2*i)+1) * Cw_q(i, k) * (sum(j=0, ra(N+2)+1, prod(u=1, j, sum(v=0, (N-(tri(u)*k))/(k+2), t^((2*v)+1) * q^(((2*v)+1)*u*k) * Cw_q(v, k)))))^2)}
A_q(k, N) = {my(q='q+O('q^N), Aqt = A_qt(k, N), Aq = 1); for(i=1, poldegree(Aqt, t), Aq += polcoef(Aqt, i, t)/(1-q^i)); Aq}
A214247_array(maxrow, maxcolumn) = {my(m=concat([1], vector(maxrow, n, numdiv(n)))~); for(k=1, maxcolumn, m = matconcat([m, Vec(A_q(k, maxrow))~])); m}
A214247_array(10, 10) \\ John Tyler Rascoe, Oct 15 2024

Crossrefs

Columns k=0-2 give: A000005, A173258, A214254.

Rows n=0, 1 and main diagonal give: A000012.

Cf. A000108, A214246, A214248, A214249, A214257, A214258, A214268, A214269, A227543.

Sequence in context: A143654 A170981 A161096 * A211987 A165983 A300719

Adjacent sequences: A214244 A214245 A214246 * A214248 A214249 A214250

Keyword

nonn,tabl,look

Author

Alois P. Heinz, Jul 08 2012

Status

approved