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
Rows n=0, 1 and main diagonal give: A000012.
AUTHOR
Alois P. Heinz, Jul 08 2012
STATUS
approved