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

0,5

Links

Table of n, a(n) for n=0..44.

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.

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)

Example

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;
  ...

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 *)

Prog

(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(""))

Crossrefs

Cf. A003319, A004208, A111528.

Sequence in context: A021830 A247686 A352369 * A111596 A271703 A276922

Adjacent sequences: A111181 A111182 A111183 * A111185 A111186 A111187

Keyword

nonn,tabl

Author

Philippe Deléham and Paul D. Hanna, Oct 16 2005

Status

approved