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A138464 Triangle read by rows: T(n,k) = number of forests on n labeled nodes with k edges (n>=1, 0<=k<=n-1). 17
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1275

FORMULA

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)

E.g.f.: exp( Sum_{n >= 1} n^(n-2)*t^(n-1)*x^n/n! ). - Peter Bala, Nov 08 2015

EXAMPLE

Triangle begins:

1;

1,  1;

1,  3,  3;

1,  6, 15,  16;

1, 10, 45, 110, 125;

MAPLE

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

MATHEMATICA

t[0, 0] = 1; t[n_ /; n >= 1, k_] /; (0 <= k <= n-1) := t[n, k] = Sum[(i+1)^(i-1)*Binomial[n-1, i]*t[n-i-1, k-i], {i, 0, k}]; t[_, _] = 0; Table[t[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* Jean-Fran├žois Alcover, Jan 14 2014, after Peter Bala *)

CROSSREFS

Row sums give A001858. Rightmost diagonal gives A000272. Cf. A136605.

Rows reflected give A105599. - Alois P. Heinz, Oct 28 2011

Cf. A088956.

Lower diagonals give: A083483, A239910, A240681, A240682, A240683, A240684, A240685, A240686, A240687. - Alois P. Heinz, Apr 11 2014

Sequence in context: A240439 A243211 A199034 * A117279 A234251 A049323

Adjacent sequences:  A138461 A138462 A138463 * A138465 A138466 A138467

KEYWORD

nonn,tabl,look

AUTHOR

N. J. A. Sloane, May 09 2008

EXTENSIONS

More terms from Alois P. Heinz, Sep 02 2008

STATUS

approved

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Last modified December 10 11:53 EST 2016. Contains 279001 sequences.