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A038719
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Triangle T(n,k) (0<=k<=n) giving number of chains of length k in partially ordered set formed from subsets of n-set by inclusion.
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4
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1, 2, 1, 4, 5, 2, 8, 19, 18, 6, 16, 65, 110, 84, 24, 32, 211, 570, 750, 480, 120, 64, 665, 2702, 5460, 5880, 3240, 720, 128, 2059, 12138, 35406, 57120, 52080, 25200, 5040, 256, 6305, 52670, 213444, 484344, 650160, 514080, 221760, 40320, 512, 19171
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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REFERENCES
| R. B. Nelsen and H. Schmidt, Jr., Chains in power sets, Math. Mag., 64 (1991), 23-31.
L. Bartlomiejczyk and J. Drewniak, A characterization of sets and operations invariant under bijections, Aequationes Mathematicae 68 (2004), pp. 1-9.
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LINKS
| Alois P. Heinz, Rows n = 0..140, flattened
Index entries for sequences related to posets
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FORMULA
| T(n, k) = Sum_{j=0..k} (-1)^j*C(k, j)*(k+2-j)^n.
T(n+1, k) = k*T(n, k-1) + (k+2)*T(n, k), T(0,0) = 1, T(0,k) = 0 for k>0.
E.g.f.: exp(2*x)/(1+y*(1-exp(x))). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 21 2003
A038719 as a lower triangular matrix is the binomial transform of A028246. - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 15 2005
Binomial transform of n-th row = 2^n + 3^n + 4^n...; e.g. binomial transform of [8, 19, 18, 6] = 2^3 + 3^3 + 4^3 + 5^3... = 8, 27, 64, 125... - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 15 2005
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EXAMPLE
| 1;
2, 1;
4, 5, 2;
8, 19, 18, 6;
16, 65, 110, 84, 24;
...
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MAPLE
| T:= proc(n, k) option remember;
`if` (n=0, `if`(k=0, 1, 0), k*T(n-1, k-1) +(k+2)*T(n-1, k))
end:
seq (seq (T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Aug 2 2011
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MATHEMATICA
| t[n_, k_] := Sum[ (-1)^(k-i)*Binomial[k, i]*(2+i)^n, {i, 0, k}]; Flatten[ Table[ t[n, k], {n, 0, 9}, {k, 0, n}]] (* From Jean-François Alcover, after Pari *)
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PROG
| (PARI) T(n, k)=sum(i=0, k, (-1)^(k-i)*binomial(k, i)*(2+i)^n)
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CROSSREFS
| Row sums give A007047. Columns give A000079, A001047, A038721. Next-to-last diagonal gives A038720.
Cf. A028246.
Sequence in context: A051176 A145064 A144332 * A125751 A099492 A144203
Adjacent sequences: A038716 A038717 A038718 * A038720 A038721 A038722
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KEYWORD
| nonn,easy,nice,tabl
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), May 02 2000
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EXTENSIONS
| More terms from Larry Reeves (larryr(AT)acm.org), May 09 2000
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