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A138464 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.

%I A138464 #61 Jun 20 2021 04:43:24
%S A138464 1,1,1,1,3,3,1,6,15,16,1,10,45,110,125,1,15,105,435,1080,1296,1,21,
%T A138464 210,1295,5250,13377,16807,1,28,378,3220,18865,76608,200704,262144,1,
%U A138464 36,630,7056,55755,320544,1316574,3542940,4782969,1,45,990,14070,143325,1092105,6258000,26100000,72000000,100000000
%N A138464 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.
%C A138464 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
%H A138464 Alois P. Heinz, <a href="/A138464/b138464.txt">Rows n = 1..141, flattened</a>
%H A138464 Federico Ardila, Matthias Beck, and Jodi McWhirter, <a href="https://doi.org/10.18257/raccefyn.1189">The arithmetic of Coxeter permutahedra</a>, Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. 44(173):1152-1166, 2020.
%H A138464 John Riordan and N. J. A. Sloane, <a href="/A003471/a003471_1.pdf">Correspondence, 1974</a>
%F A138464 From _Peter Bala_, Aug 14 2012: (Start)
%F A138464 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.
%F A138464 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.
%F A138464 The production matrix for the row polynomials of the triangle is obtained from A088956 and starts:
%F A138464     1    t
%F A138464     1    1    t
%F A138464     3    2    1    t
%F A138464    16    9    3    1    t
%F A138464   125   64   18    4    1   t
%F A138464 (End)
%F A138464 E.g.f.: exp( Sum_{n >= 1} n^(n-2)*t^(n-1)*x^n/n! ). - _Peter Bala_, Nov 08 2015
%F A138464 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]
%e A138464 Triangle begins:
%e A138464 [1]  1;
%e A138464 [2]  1,  1;
%e A138464 [3]  1,  3,   3;
%e A138464 [4]  1,  6,  15,   16;
%e A138464 [5]  1, 10,  45,  110,  125;
%e A138464 [6]  1, 15, 105,  435, 1080,  1296;
%e A138464 [7]  1, 21, 210, 1295, 5250, 13377, 16807;
%p A138464 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
%p A138464 alias(W = LambertW): EhrA := exp(-W(-t*x)/t - W(-t*x)^2/(2*t)):
%p A138464 ser := series(EhrA, x, 12): cx := n -> n!*coeff(ser, x, n):
%p A138464 T := n -> seq(coeff(cx(n), t, k), k=0..n-1):
%p A138464 seq(T(n), n = 1..10); # _Peter Luschny_, Apr 30 2021
%t A138464 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_ *)
%t A138464 gf := E^(-(ProductLog[-(t x)] (2 + ProductLog[-(t x)]))/(2 t));
%t A138464 ser := Series[gf, {x, 0, 12}]; cx[n_] := n! Coefficient[ser, x, n];
%t A138464 Table[CoefficientList[cx[n], t], {n, 1, 10}] // Flatten  (* _Peter Luschny_, May 01 2021 *)
%Y A138464 Row sums give A001858. Rightmost diagonal gives A000272. Cf. A136605.
%Y A138464 Rows reflected give A105599. - _Alois P. Heinz_, Oct 28 2011
%Y A138464 Cf. A088956.
%Y A138464 Lower diagonals give: A083483, A239910, A240681, A240682, A240683, A240684, A240685, A240686, A240687. - _Alois P. Heinz_, Apr 11 2014
%Y A138464 T(2n,n) gives A302112.
%Y A138464 For Ehrhart polynomials of integral Coxeter permutahedra of classical type cf. this sequence (type A), A343805 (type B), A343806 (type C), A343807 (type D).
%K A138464 nonn,tabl,look
%O A138464 1,5
%A A138464 _N. J. A. Sloane_, May 09 2008
%E A138464 More terms from _Alois P. Heinz_, Sep 02 2008