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A138464
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Triangle read by rows: T(n,k) = number of forests on n labeled nodes with k edges (n>=1, 0<=k<=n-1).
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8
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1, 1, 1, 1, 3, 3, 1, 6, 15, 16, 1, 10, 45, 110, 125, 1, 15, 105, 435, 1080, 1296, 1, 21, 210, 1295, 5250, 13377, 16807, 1, 28, 378, 3220, 18865, 76608, 200704, 262144, 1, 36, 630, 7056, 55755, 320544, 1316574, 3542940, 4782969, 1, 45, 990, 14070, 143325
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OFFSET
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1,5
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 1..1275
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FORMULA
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From Peter Bala, Aug 14 2012: (Start)
T(n+1,k) = sum_{i=0..k} (i+1)^(i-1)*binomial(n,i)*T(n-i,k-i) with T(0,0)=1.
Recurrence equation for row polynomials R(n,t): R(n,t) = sum_{k=0..n-1} (k+1)^(k-1)*binomial(n-1,k)*t^k*R(n-k-1,t) with R(0,t) = R(1,t) = 1.
The production matrix for the row polynomials of the triangle is obtained from A088956:
/...1....t
|...1....1....t
|...3....2....1...t
|..16....9....3...1....t
|.125...64...18...4....1....t
|...
(End)
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EXAMPLE
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Triangle begins:
1;
1, 1;
1, 3, 3;
1, 6, 15, 16;
1, 10, 45, 110, 125;
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MAPLE
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T:= proc(n) option remember; if n=0 then 0 else T(n-1) +n^(n-1) *x^n/n! fi end: TT:= proc(n) option remember; expand (T(n) -T(n)^2/2) end: f:= proc(k) option remember; if k=0 then 1 else unapply (f(k-1)(x) +x^k/k!, x) fi end: A:= proc(n, k) option remember; series(f(k)(TT(n)), x, n+1) end: aa:= (n, k)-> coeff (A(n, k), x, n) *n!: a:= (n, k)-> aa(n, n-k) -aa(n, n-k-1): seq (seq (a(n, k), k=0..n-1), n=1..10); # Alois P. Heinz, Sep 02 2008
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CROSSREFS
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Row sums give A001858. Rightmost diagonal gives A000272. Cf. A136605.
Rows reflected give A105599. - Alois P. Heinz, Oct 28 2011
Cf. A088956.
Sequence in context: A193560 A001498 A199034 * A117279 A049323 A084144
Adjacent sequences: A138461 A138462 A138463 * A138465 A138466 A138467
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KEYWORD
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nonn,tabl
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AUTHOR
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N. J. A. Sloane, May 09 2008
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EXTENSIONS
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More terms from Alois P. Heinz, Sep 02 2008
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STATUS
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approved
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