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A071919
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Number of monotone nondecreasing functions [n]->[m] for n>=0, m>=0, read by antidiagonals.
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11
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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
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OFFSET
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0,8
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COMMENTS
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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 DELEHAM, 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 DELEHAM, Nov 21 2006
A129186*A007318 as infinite lower triangular matrices. [From Philippe DELEHAM, 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}. [From Geoffrey Critzer, Jun 03 2009]
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]
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REFERENCES
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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.
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LINKS
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Table of n, a(n) for n=0..83.
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]
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FORMULA
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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.
T(n, k)=1 if (n, k)=(0, 0), a(n, k)=binomial(n+k-1, n) otherwise. - 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
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MATHEMATICA
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Table[Table[Binomial[m - 1 + n, n], {m, 0, 10}], {n, 0, 10}] // Grid [From Geoffrey Critzer, Jun 03 2009]
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PROG
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(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)=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 */
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CROSSREFS
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A000110 [From Gary W. Adamson, Jan 02 2009]
Sequence in context: A213889 A110555 * A097805 A167763 A127839 A017827
Adjacent sequences: A071916 A071917 A071918 * A071920 A071921 A071922
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KEYWORD
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nonn,easy,tabl
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AUTHOR
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Michele Dondi (bik.mido(AT)tiscalinet.it), Jun 14, 2002
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EXTENSIONS
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More terms from Reinhard Zumkeller, Jul 27 2005
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STATUS
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approved
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