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A007799
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Triangle read by rows: Whitney numbers of second kind a(n,k), n >= 1, k >= 0, for the star poset.
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1
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1, 1, 1, 1, 2, 2, 1, 1, 3, 6, 9, 5, 1, 4, 12, 30, 44, 26, 3, 1, 5, 20, 70, 170, 250, 169, 35, 1, 6, 30, 135, 460, 1110, 1689, 1254, 340, 15, 1, 7, 42, 231, 1015, 3430, 8379, 13083, 10408, 3409, 315, 1, 8, 56, 364, 1960, 8540, 28994, 71512, 114064, 96116, 36260
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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REFERENCES
| F. J. Portier and T. P. Vaughan, Whitney numbers of the second kind for the star poset, Europ. J. Combin., 11 (1990), 277-288.
K. Qiu and S. G. Akl, On some properties of the star graph, VLSI Design, Vol. 2, No. 4 (1995), 389-396.
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FORMULA
| a(n, 0)=1, a(n, 1)=n-1, a(n, 2)=(n-1)(n-2), a(n, k)=a(n-1, k)+(n-1)a(n-1, k-1)-(n-2)a(n-2, k-1)+(n-2)a(n-2, k-3) for k >= 3.
a(n, 0) = 1, a(n, 1) = n - 1, a(n, 2) = (n-1)(n-2); a(n, i) = (n-1)a(n-1, i-1) + sum_{j=1 ... n-2} j a(j, i-3). Let C(m, n) be the binomial coefficient m choose n. For 0 <= i <= ceil(3(n-1)/2) and n >= 1 we have sum_{k=0 ... i+1} (-1)^k C(i+1, k) a(n+i+1-k, i) = 0. For example, for i=2, we have a(n+3, 2) - 3a(n+2, 2) + 3a(n+1, 2) - a(n, 2) = 0. - Ke Qiu (kqiu(AT)brocku.ca), Feb 06 2005
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EXAMPLE
| 1; 1,1; 1,2,2,1; 1,3,6,9,5; 1,4,12,30,44,26,3; ...
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MATHEMATICA
| nmax = 9; a[n_, 0] = 1; a[n_, 1] = n-1; a[n_, 2] = (n-1)(n-2); a[n_, k_ /; k >= 2] := a[n, k] = (n-1)a[n-1, k-1] + Sum[j*a[j, k-3], {j, 1, n-2}]; Flatten[ Table[a[n, k], {n, 1, nmax}, {k, 0, Floor[3(n-1)/2]}]] (* From Jean-François Alcover, Nov 10 2011, after Ke Qiu *)
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CROSSREFS
| Sequence in context: A010048 A055870 A088459 * A122888 A092113 A045995
Adjacent sequences: A007796 A007797 A007798 * A007800 A007801 A007802
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KEYWORD
| nonn,tabf,easy,nice
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AUTHOR
| Frederick J. Portier [ fportier(AT)msmary.edu ]
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EXTENSIONS
| More terms from Larry Reeves (larryr(AT)acm.org), Mar 22 2000
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