cp's OEIS Frontend

A235595 Triangle read by rows: the triangle in A034855, with the n-th row normalized by dividing it by n.

%I A235595 #44 Nov 26 2021 16:36:30
%S A235595 1,1,2,1,9,6,1,40,60,24,1,195,560,420,120,1,1056,5550,6240,3240,720,1,
%T A235595 6321,59472,94710,68880,27720,5040,1,41392,692440,1527456,1426320,
%U A235595 792960,262080,40320,1,293607,8753040,26418168,30560544,21213360,9676800,2721600,362880,1,2237920,119723130,490458240,691331760,570810240,323114400,125798400,30844800,3628800
%N A235595 Triangle read by rows: the triangle in A034855, with the n-th row normalized by dividing it by n.
%C A235595 T(n,k) is the number of forests of labeled rooted trees with n nodes and height k Cf. A210725.  Equivalently, T(n,k) is the number of nilpotent partial functions on [n] with index k+1. - _Geoffrey Critzer_, Nov 26 2021
%H A235595 Alois P. Heinz, <a href="/A235595/b235595.txt">Rows n = 2..142, flattened</a>
%F A235595 A234953(n) = Sum_{k=1..n} k*T(n,k).
%e A235595 Triangle begins:
%e A235595 1.
%e A235595 1, 2,
%e A235595 1, 9, 6,
%e A235595 1, 40, 60, 24,
%e A235595 1, 195, 560, 420, 120,
%e A235595 1, 1056, 5550, 6240, 3240, 720,
%e A235595 1, 6321, 59472, 94710, 68880, 27720, 5040,
%e A235595 1, 41392, 692440, 1527456,1426320, 792960, 262080, 40320,
%e A235595 1, 293607, 8753040, 26418168, 30560544, 21213360, 9676800, 2721600, 362880,
%e A235595 ...
%p A235595 b:= proc(n, h) option remember; `if`(min(n, h)=0, 1, add(
%p A235595       binomial(n-1, j-1)*j*b(j-1, h-1)*b(n-j, h), j=1..n))
%p A235595     end:
%p A235595 T:= (n,k)-> b(n-1, k-1)-b(n-1, k-2):
%p A235595 seq(seq(T(n, d), d=1..n-1), n=2..12);  # _Alois P. Heinz_, Aug 21 2017
%t A235595 gf[k_] := gf[k] = If[k == 0, x, x*E^gf[k-1]]; a[n_, k_] := n!*Coefficient[Series[gf[k], {x, 0, n+1}], x, n]; t[n_, k_] := (a[n, k] - a[n, k-1])/n; Table[t[n, k], {n, 2, 11}, {k, 1, n-1}] // Flatten (* _Jean-François Alcover_, Mar 18 2014, after _Alois P. Heinz_ *)
%o A235595 (Python)
%o A235595 from sympy import binomial
%o A235595 from sympy.core.cache import cacheit
%o A235595 @cacheit
%o A235595 def b(n, h): return 1 if min(n, h)==0 else sum([binomial(n - 1, j - 1)*j*b(j - 1, h - 1)*b(n - j, h) for j in range(1, n + 1)])
%o A235595 def T(n, k): return b(n - 1, k - 1) - b(n - 1, k - 2)
%o A235595 for n in range(2, 13): print([T(n, d) for d in  range(1, n)]) # _Indranil Ghosh_, Aug 26 2017, after Maple code
%Y A235595 Cf. A000435, A034855, A001854, A210725, A234953, A235596, A236396.
%Y A235595 Row sums give A000272.
%K A235595 nonn,tabl
%O A235595 2,3
%A A235595 _N. J. A. Sloane_, Jan 14 2014