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A106195
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Riordan array (1/(1-2x),x(1-x)/(1-2x)).
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2
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1, 2, 1, 4, 3, 1, 8, 8, 4, 1, 16, 20, 13, 5, 1, 32, 48, 38, 19, 6, 1, 64, 112, 104, 63, 26, 7, 1, 128, 256, 272, 192, 96, 34, 8, 1, 256, 576, 688, 552, 321, 138, 43, 9, 1, 512, 1280, 1696, 1520, 1002, 501, 190, 53, 10, 1, 1024, 2816, 4096, 4048, 2972, 1683, 743, 253, 64, 11
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Extract antidiagonals from the product P * A, where P = the infinite lower triangular Pascal's triangle matrix; and A = the Pascal's triangle array:
1, 1, 1, 1...
1, 2, 3, 4...
1, 3, 6, 10..
1, 4, 10, 20..
...
Row sums are F(2n+2). Diagonal sums are A006054(n+2). Row sums of inverse are A105523. Product of Pascal triangle A007318 and A046854.
A106195 with an appended column of ones = A055587. Alternatively, k-th column (k=0, 1, 2) is the binomial transform of bin(n, k).
T(n,k) is the number of ideals in the fence Z(2n) having k elements of rank 1. [Emanuele Munarini, Mar 22 2011]
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REFERENCES
| E. Munarini, N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.
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FORMULA
| Number triangle T(n,k)=sum{j=0..n, C(n-k,n-j)*C(j,k)}
Contribution from Emanuele Munarini, Mar 22 2011: (Start)
T(n,k)=sum(binomial(k,i)*binomial(n-k,i)*2^(n-k-i),i=0..n-k)
T(n,k)=sum(binomial(k,i)*binomial(n-i,k)*(-1)^i*2^(n-k-i),i=0..n-k)
Recurrence: T(n+2,k+1)=2*T(n+1,k+1)+T(n+1,k)-T(n,k) (End)
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EXAMPLE
| Triangle begins
1,
2, 1,
4, 3, 1,
8, 8, 4, 1,
16, 20, 13, 5, 1,
32, 48, 38, 19, 6, 1,
64, 112, 104, 63, 26, 7, 1
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PROG
| (Maxima) create_list(sum(binomial(i, k)*binomial(n-k, n-i), i, 0, n), n, 0, 8, k, 0, n); [Emanuele Munarini, Mar 22 2011]
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CROSSREFS
| Column 0 = 1, 2, 4...; (binomial transform of 1, 1, 1...); column 1 = 1, 3, 8, 20...(binomial transform of 1, 2, 3...); column 2: 1, 4, 13, 38...= binomial transform of bin(n, 2): 1, 3, 6...
Cf. A055587, A007318, A001792, A002620, A049612.
Sequence in context: A103316 A140069 A105851 * A051129 A067410 A188403
Adjacent sequences: A106192 A106193 A106194 * A106196 A106197 A106198
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 24 2005; Paul Barry (pbarry(AT)wit.ie), May 21 2006
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Apr 09 2007, merging two sequences submitted independently by Gary W. Adamson and Paul Barry
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