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A143395
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Triangle T(n,k) = number of forests of k labeled rooted trees of height at most 1, with n labels, where any root may contain >= 1 labels, n>=0, 0<=k<=n.
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8
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1, 0, 1, 0, 3, 1, 0, 7, 9, 1, 0, 15, 55, 18, 1, 0, 31, 285, 205, 30, 1, 0, 63, 1351, 1890, 545, 45, 1, 0, 127, 6069, 15421, 7770, 1190, 63, 1, 0, 255, 26335, 116298, 95781, 24150, 2282, 84, 1, 0, 511, 111645, 830845, 1071630, 416451, 62370, 3990, 108, 1, 0, 1023
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OFFSET
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0,5
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COMMENTS
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This is the Sheffer triangle (1,exp(x)*(exp(x)-1)) (Jobotinsky type). See the e.g.f. given by V. Jovovic below, and the W. Lang link under A006232 (second part) for general Sheffer remarks and the conversion to the umbral notation of S. Roman's book. - Wolfdieter Lang, Oct 08 2011
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LINKS
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Alois P. Heinz, Rows n = 0..140, flattened
Index entries for sequences related to rooted trees
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FORMULA
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G.f. of column k: x^k/Prod_{t=k..2*k}(1-t*x).
T(n,k) = Sum_{t=k..n} C(n,t) * stirling2(t,k) * k^(n-t).
E.g.f.: exp(y*exp(x)*(exp(x)-1)). - Vladeta Jovovic, Dec 08 2008
T(n,k) = Sum_{m=0..k} stirling2(n,k+m)*(k+m)!/(m!*(k-m)!). - Vladimir Kruchinin, Apr 06 2011
Let P be Pascal's triangle A007318. The first column of the array exp(t*(P^2-P)) gives the row generating polynomials of this triangle.
The row polynomials R(n,t) satisfy the recurrence R(n+1,t) = t*(sum {k = 0..n} (2^(k+1)-1)*C(n,k)*R(n-k,t)) with R(0,t) = 1. For example, the row 4 polynomial R(4,t) = 15*t+55*t^2+18*t^3+t^4 = t*((7*t+9*t^2+t^3) + 3*3*(3*t+t^2) + 7*3*t + 15*1). - Peter Bala, Oct 12 2011
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EXAMPLE
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T(3,2) = 9: {1}{2}<-3, {1}{3}<-2, {1}{2,3}, {2}{1}<-3, {2}{3}<-1, {2}{1,3}, {3}{1}<-2, {3}{2}<-1, {3}{1,2}.
Triangle begins:
1;
0, 1;
0, 3, 1;
0, 7, 9, 1;
0, 15, 55, 18, 1;
0, 31, 285, 205, 30, 1;
0, 63, 1351, 1890, 545, 45, 1;
0, 127, 6069, 15421, 7770, 1190, 63, 1;
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MAPLE
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with(combinat): T:= (n, k)-> add(binomial(n, t)* stirling2(t, k)* k^(n-t), t=k..n): seq(seq(T(n, k), k=0..n), n=0..11);
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CROSSREFS
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Columns k=0-9: A000007, A000225, A016269(n-2), A028025(n-3), A143399, A143400, A143401, A143402, A143403, A143404. Diagonal: A000012. See also A048993, A008277, A007318, A143405 for row sums.
Sequence in context: A111246 A206306 A178124 * A090536 A187557 A052420
Adjacent sequences: A143392 A143393 A143394 * A143396 A143397 A143398
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KEYWORD
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nonn,tabl,changed
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AUTHOR
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Alois P. Heinz, Aug 12 2008
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STATUS
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approved
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