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

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, 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, 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

Offset: 0

Author

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

Keywords

• nonn
• easy
• tabl

Comments

Sometimes called a Riordan array.

Number of different partial sums of 1 + [2,3] + [3,4] + [4,5] + ... - Jon Perry, Jan 01 2004

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

T(n,k)=abs(A110555(n,k)), A110555(n,k)=T(n,k)*(-1)^k. - Reinhard Zumkeller, Jul 27 2005

(1,0)-Pascal triangle. - Philippe Deléham, Nov 21 2006

A129186*A007318 as infinite lower triangular matrices. - Philippe Deléham, Mar 07 2009

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

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

For a closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - Boris Putievskiy, Aug 19 2013

For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 09 2013

Examples

			   1,    1,    1,    1,    1,    1,    1,    1,    1, ...
   0,    1,    2,    3,    4,    5,    6,    7,    8, ...
   0,    1,    3,    6,   10,   15,   21,   28,   36, ...
   0,    1,    4,   10,   20,   35,   56,   84,  120, ...
   0,    1,    5,   15,   35,   70,  126,  210,  330, ...
   0,    1,    6,   21,   56,  126,  252,  462,  792, ...
   0,    1,    7,   28,   84,  210,  462,  924, 1716, ...
   0,    1,    8,   36,  120,  330,  792, 1716, 3432, ...
   0,    1,    9,   45,  165,  495, 1287, 3003, 6435, ...
		

Links

• G. C. Greubel, Table of n, a(n) for the first 101 antidiagonals, flattened
• 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.
• Wolfdieter Lang, Simple proofs of some facts related to the Bell sequence and triangles A007318 (Pascal) and A071919 (enlarged Pascal). [From _Wolfdieter Lang_, Jun 23 2010]

Crossrefs

Cf. A000110, A007318, A228196, A228576.

Columns are A000007, A000012, A000027, A000217, A000292, A000332, A000389, ...

Main diagonal gives A088218.

Programs

Maple

A:= (n, m)-> binomial(n+m-1, n):
seq(seq(A(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Jan 13 2017

Mathematica

Table[Table[Binomial[m - 1 + n, n], {m, 0, 10}], {n, 0, 10}] // Grid (* Geoffrey Critzer, Jun 03 2009 *)
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 *)

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

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 */

Formula

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.

A(n,k) = binomial(n+k-1,n). - Reinhard Zumkeller, Jul 27 2005

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

G.f. of the triangular interpretation: (-1+x*y)/(-1+x*y+x). - R. J. Mathar, Aug 11 2015