0,8

Triangle T(n,k) is [0,1,1,2,2,3,3,4,4,...] DELTA [1,0,0,0,0,0.....] = A001057 DELTA [1,0,0,0,.]

T(n,k) = number of permutations on [n] that (i) contain a 132 pattern only as part of a 4132 pattern and (ii) start with n+1-k. For example, for n>=1, T(n,1) = (n-1)! counts all (n-1)! permutations on [n] that start with n: either they avoid 132 altogether or the initial entry serves as the "4" in a 4132 pattern, and T(4,3) = 3 counts 2134, 2314, 2341. - David Callan (callan(AT)stat.wisc.edu), Jul 20 2005

T(n,k) is the number of permutations on [n] that (i) contain a (scattered) 342 pattern only as part of a 1342 pattern, and (ii) contain 1 in position k. For example, T(4,3) counts 3214, 4213, 4312. (It does not count, say, 2314 because 231 forms an offending 342 pattern.) - David Callan (callan(AT)stat.wisc.edu), Jul 20 2005

This triangle * [1,2,3,...] = A134378: (1, 2, 5, 14, 44, 158, 663,...) = row sums of triangle A134379. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 22 2007

David Callan, A Combinatorial Interpretation of the Eigensequence for Composition, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.4.

David Callan, A combinatorial interpretation of the eigensequence for composition

# The operator DELTA takes two sequences r = (r_0, r_1, ...), s = (s_0, s_1, ...) and produces a triangle T(n, k), 0 <= k <= n, as follows:

Let q(k) = x*r_k + y*s_k for k >= 0; let P(n, k) (n >= 0, k >= -1) be defined recursively by P(0, k) = 1 for k >= 0; P(n, -1) = 0 for n >= 1; P(n, k) = P(n, k-1) + q(k)*P(n-1, k+1) for n >= 1, k >= 0.

Then P(n, k) is a homogeneous polynomial in x and y of degree n, and T(n, k) = coefficient of x^(n-k)*y^k in P(n, 0).

T(m+n, m)= Sum_{k=0..n} A090238(n, k)*binomial(m, k).

G.f. for column k: Sum_{n>=0} T(k+n, k)*x^n = (Sum_{n>=0} n!*x^n )^k.

For k>0, T(n+k, k) = Sum_{a_1 + a_2 + .. + a_k = n} (a_1)!*(a_2)!*..*(a_k)!; a_i>=0, n>=0.

{1}, {0, 1}, {0, 1, 1}, {0, 2, 2, 1}, {0, 6, 5, 3, 1}, {0, 24, 16, 9, 4, 1}, ...

DELTA := proc(r, s, n) local T, x, y, q, P, i, j, k, t1; T := array(0..n, 0..n);

for i from 0 to n do q[i] := x*r[i+1]+y*s[i+1]; od: for k from 0 to n do P[0, k] := 1; od: for i from 0 to n do P[i, -1] := 0; od:

for i from 1 to n do for k from 0 to n do P[i, k] := sort(expand(P[i, k-1] + q[k]*P[i-1, k+1])); od: od:

for i from 0 to n do t1 := P[i, 0]; for j from 0 to i do T[i, j] := coeff(coeff(t1, x, i-j), y, j); od: lprint( seq(T[i, j], j=0..i) ); od: end;

# To produce the current triangle: s3 := n->floor((n+1)/2); s4 := n->if n = 0 then 1 else 0; fi; r := [seq(s3(i), i= 0..40)]; s := [seq(s4(i), i=0..40)]; DELTA(r, s, 20);

T(n,k) = sum(j>=0, A075834(j)*T(n-1, k+j-1)), T(k,k) = 1, T(k+1,k) = A001477(k), T(k+2,k) = A000096(k), T(n+1,1)= A000142(n), T(n+2,2) = A003149(n).

Cf. A051295 (row sums).

Diagonals : A000007, A000142, A003149, A000012, A001477, A000096, A092286, A090386, A090391, A090392, A090393, A090394.

Cf. A090238.

Cf. A134378, A134379.

Adjacent sequences: A084935 A084936 A084937 this_sequence A084939 A084940 A084941

Sequence in context: A064045 A110314 A130167 this_sequence A135898 A131182 A093729

nonn,tabl

DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jul 16 2003