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%I #64 Jan 05 2022 06:13:49
%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 126,126,84,36,9,1,0,1,10,45,120,210,252,210,120,45,10,1,0,1,11,55,165,330,462,462,330,165,55,11,1,0
%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 Deléham_, 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 Deléham_, Nov 21 2006
%C A129186*A007318 as infinite lower triangular matrices. - _Philippe Deléham_, 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}. - _Geoffrey Critzer_, Jun 03 2009
%C Sum_{k=0..n} T(n,k)*B(k) = B(n), n>=0, with the Bell numbers B(n):=A000110(n) (eigensequence). See, e.g., the W. Lang link, Corollary 4. - _Wolfdieter Lang_, Jun 23 2010
%C For a closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - _Boris Putievskiy_, Aug 19 2013
%C For a closed-form formula for generalized Pascal's triangle see A228576. - _Boris Putievskiy_, Sep 09 2013
%H G. C. Greubel, <a href="/A071919/b071919.txt">Table of n, a(n) for the first 101 antidiagonals, flattened</a>
%H D. Merlini, F. Uncini and M. C. Verri, <a href="https://www.emis.de/journals/INTEGERS/papers/e23/e23.Abstract.html">A unified approach to the study of general and palindromic compositions</a>, 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 Limit_{k->infinity} 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 A(n,k) = binomial(n+k-1,n). - _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
%F G.f. of the triangular interpretation: (-1+x*y)/(-1+x*y+x). - _R. J. Mathar_, Aug 11 2015
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 3, 4, 5, 6, 7, 8, ...
%e 0, 1, 3, 6, 10, 15, 21, 28, 36, ...
%e 0, 1, 4, 10, 20, 35, 56, 84, 120, ...
%e 0, 1, 5, 15, 35, 70, 126, 210, 330, ...
%e 0, 1, 6, 21, 56, 126, 252, 462, 792, ...
%e 0, 1, 7, 28, 84, 210, 462, 924, 1716, ...
%e 0, 1, 8, 36, 120, 330, 792, 1716, 3432, ...
%e 0, 1, 9, 45, 165, 495, 1287, 3003, 6435, ...
%p A:= (n, m)-> binomial(n+m-1, n):
%p seq(seq(A(n, d-n), n=0..d), d=0..14); # _Alois P. Heinz_, Jan 13 2017
%t Table[Table[Binomial[m - 1 + n, n], {m, 0, 10}], {n, 0, 10}] // Grid (* _Geoffrey Critzer_, Jun 03 2009 *)
%t a[n_, m_] := Binomial[m - 1 + n, n]; Table[Table[a[n, m - n], {n, 0, m}], {m, 0, 10}] // Flatten (* _G. C. Greubel_, Nov 22 2017 *)
%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) = my(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 Cf. A000110, A007318, A228196, A228576.
%Y Columns are A000007, A000012, A000027, A000217, A000292, A000332, A000389, ...
%Y Main diagonal gives A088218.
%K nonn,easy,tabl
%O 0,8
%A Michele Dondi (bik.mido(AT)tiscalinet.it), Jun 14 2002
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