%I A103371
%S A103371 1,2,1,3,6,1,4,18,12,1,5,40,60,20,1,6,75,200,150,30,1,7,126,525,700,315,
%T A103371 42,1,8,196,1176,2450,1960,588,56,1,9,288,2352,7056,8820,4704,1008,72,
               1,
%U A103371 10,405,4320,17640,31752,26460,10080,1620,90,1,11,550,7425,39600,97020
%N A103371 Number triangle T(n,k)=C(n,n-k)C(n+1,n-k).
%C A103371 Columns include A000027, A002411, A004302. Row sums are C(2n+1,n+1) or 
               A001700.
%C A103371 T(n-1,k-1) is the number of possibilities to put n identical objects 
               into k of alltogether n distinguishable boxes. See the partition 
               array A035206 from which this triangle arises after summing over 
               all entries related to partitions with fixed part number k.
%C A103371 T(n, k) is also the number of order-preserving full transformations (of 
               an n-chain) of height k (height(alpha) = |Im(alpha)|). [From A. Umar 
               (aumarh(AT)squ.edu.om), Oct 02 2008]
%D A103371 Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving 
               full transformations. Semigroup Forum 72 (2006), 51-62. [From A. 
               Umar (aumarh(AT)squ.edu.om), Oct 02 2008]
%F A103371 Number triangle T(n, k)=C(n, n-k)C(n+1, n-k)=C(n, k)C(n+1, k+1); Column 
               k of this triangle has g.f. sum{j=0..k, C(k, j)C(k+1, j)x^(k+j)}/
               (1-x)^(2k+2); coefficients of the numerators are the rows of the 
               reverse triangle C(n, k)C(n+1, k).
%F A103371 T(n, k)=C(n, k)*sum{j=0..n, C(n-j, k), j, 0, n-k}; - Paul Barry (pbarry(AT)wit.ie), 
               Jan 12 2006
%F A103371 T(n,k)= (n+1-k)*N(n+1,k+1), with N(n,k):=A001263(n,k), the Narayana triangle 
               (with offset [1,1)]
%F A103371 O.g.f.: ((1-(1-y)*x)/sqrt((1-(1+y)*x)^2-4*x^2*y) -1)/2, (from o.g.f. 
               of A001263, Narayana triangle). W. Lang, Nov 13 2007.
%F A103371 Matrix product of A007318 and A122899. O.g.f. for row n: (1-x)^n*P(n,
               1,0,(1+x)/(1-x)) = (1-x)^(n-1)*(Legendre_P(n,x) - Legendre_P(n+1,
               x)), where P(n,a,b,x) denotes the Jacobi polynomial. O.g.f. for column 
               k: x^k/(1-x)^(k+2)*P(k,0,1,(1+x)/(1-x)). Compare with A008459. - 
               Peter Bala (pbala(AT)toucansurf.com), Jan 24 2008
%e A103371 Rows start {1}, {2,1}, {3,6,1}, {4,18,12,1},...
%Y A103371 Cf. A008459, A122899.
%Y A103371 Sequence in context: A115196 A093346 A115597 this_sequence A120257 A059298 
               A156914
%Y A103371 Adjacent sequences: A103368 A103369 A103370 this_sequence A103372 A103373 
               A103374
%K A103371 easy,nonn,tabl
%O A103371 0,2
%A A103371 Paul Barry (pbarry(AT)wit.ie), Feb 03 2005