%I A084938
%S A084938 1,0,1,0,1,1,0,2,2,1,0,6,5,3,1,0,24,16,9,4,1,0,120,64,31,14,5,1,0,
%T A084938 720,312,126,52,20,6,1,0,5040,1812,606,217,80,27,7,1,0,40320,12288,
%U A084938 3428,1040,345,116,35,8,1,0,362880,95616,22572,5768,1661,519,161,44
%N A084938 Triangle of numbers T(n,k), 0<=n, 0<=k: T(n,k)= sum(j>=0) j!*T(n-j-1, 
               k-1).
%C A084938 Triangle T(n,k) is [0,1,1,2,2,3,3,4,4,...] DELTA [1,0,0,0,0,0.....] = 
               A110654 DELTA A000007
%C A084938 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
%C A084938 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
%C A084938 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
%C A084938 Riordan array (1,xg(x)) where g(x) is the g.f. of the factorials (n!). 
               [From Paul Barry (pbarry(AT)wit.ie), Sep 25 2008]
%C A084938 Modulo 2, this sequence becomes A106344 .
%C A084938 In general, the triangle [r_0,r_1,r_2,r_3,...] DELTA [s_0,s_1,s_2,s_3,
               ...] has generating function 1/(1-(r_0*x+s_0*x*y)/(1-(r_1*x+s_1*x*y)/
               (1-(r_2*x+s_2*x*y)/1-(r_3*x+s_3*x*y)/(1-...(continued fraction).
%C A084938 Eigensequence of the triangle = A165489: (1, 1, 2, 6, 23, 105, 550, 3236,
               ...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 20 2009]
%D A084938 David Callan, A Combinatorial Interpretation of the Eigensequence for 
               Composition, Journal of Integer Sequences, Vol. 9 (2006), Article 
               06.1.4.
%H A084938 David Callan, <a href="http://front.math.ucdavis.edu/math.CO/0507169">
               A combinatorial interpretation of the eigensequence for composition</
               a>
%F A084938 # 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:
%F A084938 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.
%F A084938 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).
%F A084938 T(m+n, m)= Sum_{k=0..n} A090238(n, k)*binomial(m, k).
%F A084938 G.f. for column k: Sum_{n>=0} T(k+n, k)*x^n = (Sum_{n>=0} n!*x^n )^k.
%F A084938 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.
%F A084938 T(n,k)= Sum_{ j, j>=0}A075834(j)*T(n-1,k+j-1).
%e A084938 {1}, {0, 1}, {0, 1, 1}, {0, 2, 2, 1}, {0, 6, 5, 3, 1}, {0, 24, 16, 9, 
               4, 1}, ...
%e A084938 Contribution from Paul Barry (pbarry(AT)wit.ie), Sep 25 2008: (Start)
%e A084938 Triangle [0,1,1,2,2,4,4,5,5,....] DELTA [1,0,0,0,0,....] begins
%e A084938 1,
%e A084938 0, 1,
%e A084938 0, 1, 1,
%e A084938 0, 2, 2, 1,
%e A084938 0, 6, 5, 3, 1,
%e A084938 0, 24, 16, 9, 4, 1,
%e A084938 0, 120, 64, 31, 14, 5, 1,
%e A084938 0, 720, 312, 126, 52, 20, 6, 1,
%e A084938 0, 5040, 1812, 606, 217, 80, 27, 7, 1,
%e A084938 0, 40320, 12288, 3428, 1040, 345, 116, 35, 8, 1,
%e A084938 0, 362880, 95616, 22572, 5768, 1661, 519, 161, 44, 9, 1 (End)
%e A084938 Contribution from Paul Barry (pbarry(AT)wit.ie), May 14 2009: (Start)
%e A084938 The production matrix is
%e A084938 0, 1,
%e A084938 0, 1, 1,
%e A084938 0, 1, 1, 1,
%e A084938 0, 2, 1, 1, 1,
%e A084938 0, 7, 2, 1, 1, 1,
%e A084938 0, 34, 7, 2, 1, 1, 1,
%e A084938 0, 206, 34, 7, 2, 1, 1, 1
%e A084938 which is based on A075834. (End)
%p A084938 DELTA := proc(r,s,n) local T,x,y,q,P,i,j,k,t1; T := array(0..n,0..n);
%p A084938 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:
%p A084938 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:
%p A084938 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;
%p A084938 # 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);
%Y A084938 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); T(n+3,3)= A090595(n); T(n+4,4)= 
               A090319(n).
%Y A084938 Cf. A051295 (row sums), A090238, A134378, A134379.
%Y A084938 Diagonals : A000007, A000142, A003149, A090595, A090319 ; A000012, A001477, 
               A000096, A092286, A090386, A090391, A090392, A090393, A090394.
%Y A084938 Cf. A165489, A165490 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 
               20 2009]
%Y A084938 Sequence in context: A110314 A152882 A130167 this_sequence A135898 A131182 
               A093729
%Y A084938 Adjacent sequences: A084935 A084936 A084937 this_sequence A084939 A084940 
               A084941
%K A084938 nonn,tabl
%O A084938 0,8
%A A084938 DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jul 16 2003; corrections Dec 
               17 2008, Dec 20 2008, Feb 05 2009