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 A071919 Number of monotone nondecreasing functions [n]->[m] for n>=0, m>=0, read by antidiagonals. 11

%I

%S 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,

%T 15,6,1,0,1,7,21,35,35,21,7,1,0,1,8,28,56,70,56,28,8,1,0,1,9,36,84,

%U 126,126,84,36,9,1,0,1,10,45,120,210,252,210,120,45,10,1,0,1,11,55,165,330,462

%N Number of monotone nondecreasing functions [n]->[m] for n>=0, m>=0, read by antidiagonals.

%C Sometimes called a Riordan array.

%C Number of different partial sums of 1+[2,3]+[3,4]+[4,5]+... - _Jon Perry_, Jan 01 2004

%C 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_, Sep 05 2005

%C T(n,k)=abs(A110555(n,k)), A110555(n,k)=T(n,k)*(-1)^k. - _Reinhard Zumkeller_, Jul 27 2005

%C (1,0)-Pascal triangle . - _Philippe DELEHAM_, Nov 21 2006

%C A129186*A007318 as infinite lower triangular matrices. [From _Philippe DELEHAM_, Mar 07 2009]

%C 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_, Jun 03 2009]

%C Sum(T(n,k)*B(k),k=0..n) = B(n), n>=0, with the Bell numbers B(n):=A000110(n) (eigensequence). See e.g., the W. Lang link, Corollary 4. [From _Wolfdieter Lang_, Jun 23 2010]

%D 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.

%H Wolfdieter Lang, <a href="/A071919/a071919.pdf"> Simple proofs of some facts related to the Bell sequence and triangles A007318 (Pascal) and A071919 (enlarged Pascal).</a> [From _Wolfdieter Lang_, Jun 23 2010]

%F Lim_{k->inf.} A071919^k = (A000110,0,0,0,0,...) with the Bell numbers in the first column. For a proof see, e.g., the W. Lang link, proposition 12.

%F T(n, k)=1 if (n, k)=(0, 0), a(n, k)=binomial(n+k-1, n) otherwise. - _Reinhard Zumkeller_, Jul 27 2005

%F G.f.: 1 +x +x^3(1+x) +x^6(1+x)^2 +x^10(1+x)^3 +... . - Michael Somos Aug 20 2006

%t Table[Table[Binomial[m - 1 + n, n], {m, 0, 10}], {n, 0, 10}] // Grid [From _Geoffrey Critzer_, Jun 03 2009]

%o (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 (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 A000110 [From _Gary W. Adamson_, Jan 02 2009]

%K nonn,easy,tabl

%O 0,8

%A Michele Dondi (bik.mido(AT)tiscalinet.it), Jun 14, 2002

%E More terms from _Reinhard Zumkeller_, Jul 27 2005

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