%I A071919
%S A071919 1,1,0,1,1,0,1,2,1,0,1,3,3,1,0,1,4,6,4,1,0,1,5,10,10,5,1,0,1,6,15,20,15,
%T A071919 6,1,0,1,7,21,35,35,21,7,1,0,1,8,28,56,70,56,28,8,1,0,1,9,36,84,126,126,
%U A071919 84,36,9,1,0,1,10,45,120,210,252,210,120,45,10,1,0,1,11,55,165,330,462
%N A071919 Number of monotone nondecreasing functions [n]->[m] for n>=0, m>=0, read 
               by antidiagonals.
%C A071919 Sometimes called a Riordan array.
%C A071919 Number of different partial sums of 1+[2,3]+[3,4]+[4,5]+... - Jon Perry 
               (perry(AT)globalnet.co.uk), Jan 01 2004
%C A071919 Triangle T(n,k), 0<=k<=n, read by rows, given by [1, 0, 0, 0, 0, 0, 0, 
               0, ...] DELTA [0, 1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator 
               defined in A084938 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Sep 05 2005
%C A071919 T(n,k)=abs(A110555(n,k)), A110555(n,k)=T(n,k)*(-1)^k. - Reinhard Zumkeller 
               (reinhard.zumkeller(AT)gmail.com), Jul 27 2005
%C A071919 (1,0)-Pascal triangle . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 21 2006
%C A071919 Lim_{k->inf.} A071919^k = A000110, the Bell numbers. [From Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Jan 02 2009]
%C A071919 A129186*A007318 as infinite lower triangular matrices. [From Philippe 
               DELEHAM (kolotoko(AT)wanadoo.fr), Mar 07 2009]
%C A071919 Let n>=0 index the rows and m>=0 index the columns of this rectangular 
               array. R(n,m) is "m multichoose n", the number of multisets of length 
               n on m symbols. R(n,m)= Sum_i=0...n;R(i,m-1). The summation conditions 
               on the number of members in a size n multiset that are not the element 
               m (an arbitrary element in the set of m symbols). R(n,m)= Sum_i=1...m;
               R(n-1,i). The summation conditions on the largest element in a size 
               n multiset on {1,2,...m}. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), 
               Jun 03 2009]
%D A071919 D. Merlini, F. Uncini and M. C. Verri, A unified approach to the study 
               of general and palindromic compositions, Integers 4 (2004), A23, 
               26 pp.
%F A071919 T(n, k)=1 if (n, k)=(0, 0), a(n, k)=binomial(n+k-1, n) otherwise. - Reinhard 
               Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 27 2005
%F A071919 G.f.: 1 +x +x^3(1+x) +x^6(1+x)^2 +x^10(1+x)^3 +... . - Michael Somos 
               Aug 20 2006
%t A071919 Table[Table[Binomial[m - 1 + n, n], {m, 0, 10}], {n, 0, 10}] // Grid 
               [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Jun 03 2009]
%o A071919 (PARI) { n=20; v=vector(n); for (i=1,n,v[i]=vector(2^(i-1))); v[1][1]=1; 
               for (i=2,n, k=length(v[i-1]); for (j=1,k, v[i][j]=v[i-1][j]+i; v[i][j+k]=v[i-1][j]+i+1)); 
               c=vector(n); for (i=1,n, for (j=1,2^(i-1), if (v[i][j]<=n, c[v[i][j]]++))); 
               c } (Jon Perry)
%o A071919 (PARI) {a(n)=local(m); if(n<1, n==0, m=(sqrtint(8*n+1)-1)\2; binomial(m-1,
               n-m*(m+1)/2))} /* Michael Somos Aug 20 2006 */
%Y A071919 A000110 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 02 2009]
%Y A071919 Sequence in context: A077593 A119337 A110555 this_sequence A097805 A167763 
               A127839
%Y A071919 Adjacent sequences: A071916 A071917 A071918 this_sequence A071920 A071921 
               A071922
%K A071919 nonn,easy,tabl
%O A071919 0,8
%A A071919 Michele Dondi (bik.mido(AT)tiscalinet.it), Jun 14, 2002
%E A071919 More terms from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jul 27 2005