A111184 Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] where DELTA is the operator defined in A084938.
1, 0, 1, 0, 2, 1, 0, 6, 6, 1, 0, 24, 34, 12, 1, 0, 120, 210, 110, 20, 1, 0, 720, 1452, 974, 270, 30, 1, 0, 5040, 11256, 8946, 3248, 560, 42, 1, 0, 40320, 97296, 87504, 38338, 8792, 1036, 56, 1
Offset: 0
Examples
Rows begin: 1; 0, 1; 0, 2, 1; 0, 6, 6, 1; 0, 24, 34, 12, 1; 0, 120, 210, 110, 20, 1; 0, 720, 1452, 974, 270, 30, 1; 0, 5040, 11256, 8946, 3248, 560, 42, 1; 0, 40320, 97296, 87504, 38338, 8792, 1036, 56, 1; ...
Links
- Paul Barry, A note on number triangles that are almost their own production matrix, arXiv:1804.06801 [math.CO], 2018.
- R. Cori, Indecomposable permutations, hypermaps and labeled Dyck paths, J. Comb. Theory A 116 (2009) 1326-1343, end of Section 1.2.2.
Programs
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Mathematica
DELTA[r_, s_, m_] := Module[{p, q, t, x, y}, q[k_] := x r[[k+1]] + y s[[k+1]]; p[0, ] = 1; p[, -1] = 0; p[n_ /; n >= 1, k_ /; k >= 0] := p[n, k] = p[n, k-1] + q[k] p[n-1, k+1] // Expand; t[n_, k_] := Coefficient[p[n, 0], x^(n-k) y^k]; t[0, 0] = p[0, 0]; Table[t[n, k], {n, 0, m}, {k, 0, n}]]; DELTA[LinearRecurrence[{1, 1, -1}, {0, 2, 1}, 10], Mod[Range[10], 2], 10] // Flatten (* Jean-François Alcover, Jul 27 2018 *) -
PARI
{T(n,k)=local(A=1+x*y);for(i=1,n,A=1-x*deriv(log(1+x-x*y-x*A +x*O(x^n))));polcoeff(polcoeff(A,n,x),k,y)} /* Paul D. Hanna */ -
PARI
{T(n, k)=local(A=1+x*y); for(i=1, n, A=(1 + x^2*A')/(1 + x - x*y - x*A +x*O(x^n))); polcoeff(polcoeff(A, n, x), k, y)} /* Paul D. Hanna */ /* Print 10 Rows of the triangle: */ for(n=0,10,for(k=0,n,print1(T(n,k),","));print(""))
Formula
O.g.f. satisfies: A(x,y) = (1 + x^2*A'(x,y)) / (1+x - x*y - x*A(x,y)), where A'(x,y) = d/dx A(x,y). - Paul D. Hanna, Jul 31 2011
O.g.f. satisfies: A(x,y) = 1 - x * d/dx log(1+x - x*y - x*A(x,y)). - Paul D. Hanna, Jul 30 2011
Sum_{k=0..n} T(n, k) = A003319(n+1).
Sum_{k=0..n} T(n, k)*2^(n-k) = A004208(n).
From Mikhail Kurkov, Jul 15 2025: (Start)
Conjecture 1: Sum_{k=1..n} T(n,k)*q^(k-1) = A111528(q,n) for n > 0, q >= 0.
Conjecture 2: Sum_{k=1..n} T(n,k)*(-1/q)^(k-1) = R(n,q)/q^(n-1) for n > 0, q > 0 where log(1 + x + q*x*[Sum_{k>=1} R(k,q)*x^k]) = Sum_{k>=1} R(k,q)/k*x^k.
Conjecture 3: n-th row polynomial is x*v_n for n > 0 where we start with vector v of fixed length m with elements v_i = 1, and for i=1..m-1, j=i+1..m apply A := v_i (at the beginning of each cycle for i) and also apply A := A + v_j, v_j := (j-i+x-1)*v_j + A.
Here last conjecture provides fast and simple algorithm, that allow to compute sums in the previous conjectures by substituting x = q and x = -1/q respectively. (End)