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<=k<=n} T(n, k) = A003319(n+1).

Sum_{k, 0<=k<=n} T(n, k)*2^(n-k) = A004208(n).

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.

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