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A138464
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Triangle read by rows: T(n, k) is the number of forests on n labeled nodes with k edges. T(n, k) for n >= 1 and 0 <= k <= n-1.
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22
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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, 1092105, 6258000, 26100000, 72000000, 100000000
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OFFSET
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1,5
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COMMENTS
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The rows of the triangle give the coefficients of the Ehrhart polynomials of integral Coxeter permutahedra of type A. These polynomials count lattice points in a dilated lattice polytope. For a definition see Ardila et al. (p. 1158), the generating functions of these polynomials for the classical root systems are given in theorem 5.2 (p. 1163). - Peter Luschny, May 01 2021
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LINKS
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FORMULA
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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 and starts:
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
T(n, k) = [t^k] n! [x^n] exp(-W(-t*x)/t - W(-t*x)^2/(2*t)), where W denotes the Lambert function. - Peter Luschny, Apr 30 2021 [Typo corrected after note from Andrew Howroyd, Peter Luschny, Jun 20 2021]
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EXAMPLE
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Triangle begins:
[1] 1;
[2] 1, 1;
[3] 1, 3, 3;
[4] 1, 6, 15, 16;
[5] 1, 10, 45, 110, 125;
[6] 1, 15, 105, 435, 1080, 1296;
[7] 1, 21, 210, 1295, 5250, 13377, 16807;
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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
alias(W = LambertW): EhrA := exp(-W(-t*x)/t - W(-t*x)^2/(2*t)):
ser := series(EhrA, x, 12): cx := n -> n!*coeff(ser, x, n):
T := n -> seq(coeff(cx(n), t, k), k=0..n-1):
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MATHEMATICA
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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-François Alcover, Jan 14 2014, after Peter Bala *)
gf := E^(-(ProductLog[-(t x)] (2 + ProductLog[-(t x)]))/(2 t));
ser := Series[gf, {x, 0, 12}]; cx[n_] := n! Coefficient[ser, x, n];
Table[CoefficientList[cx[n], t], {n, 1, 10}] // Flatten (* Peter Luschny, May 01 2021 *)
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CROSSREFS
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For Ehrhart polynomials of integral Coxeter permutahedra of classical type cf. this sequence (type A), A343805 (type B), A343806 (type C), A343807 (type D).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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