OFFSET
0,8
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
(1,0)-Pascal triangle. - Philippe Deléham, Nov 21 2006
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
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]
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
EXAMPLE
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, ...
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 *)
PROG
(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 */
CROSSREFS
KEYWORD
AUTHOR
Michele Dondi (bik.mido(AT)tiscalinet.it), Jun 14 2002
STATUS
approved