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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.
12
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.
Sequence in context: A143654 A170981 A161096 * A211987 A165983 A300719
KEYWORD
nonn,tabl,look
AUTHOR
Alois P. Heinz, Jul 08 2012
STATUS
approved