A038081 Number of rooted identity trees of height n. Sets of rank n.
1, 1, 2, 12, 65520
Offset: 0
Links
- Eric Weisstein's World of Mathematics, Rank.
- Index entries for sequences related to rooted trees
This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
For n = 3, the n-th iteration of the von Neumann universe is V3 = {{}, {{}}, {{{}}}, {{},{{}}}}, which has a(3) = 11 pairs of braces.
h:= (b, k)-> `if`(k=0, 1, b^h(b, k-1)):
a:= proc(n) option remember; `if`(n=0, 1,
1+(1+a(n-1))/2*h(2, n-1))
end:
seq(a(n), n=0..5); # Alois P. Heinz, Aug 25 2017
Map[#[[1]]&,NestList[{(#[[1]]+1)*(2^#[[2]])/2+1,2^#[[2]]}&,{1,0},6]] (* Adam P. Goucher, Aug 18 2013 *)
weigh:= proc(p) proc(n) `if`(n<0,1, coeff(mul((1+x^k)^p(k), k=1..n), x,n)) end end: wsh:= p-> n-> weigh(p)(n-1): a:= wsh(n-> `if`(n>0 and n<12, [1$3,2$5,1$3][n],0)): seq(a(n), n=1..97); # Alois P. Heinz, Sep 10 2008
a = Drop[CoefficientList[ Series[x (1 + x) (1 + x^2) (1 + x^3) (1 + x^4), {x, 0, 11}], x], 1]; nn = 97; Drop[ CoefficientList[ Series[x Product[(1 + x^i)^a[[i]], {i, 1, 11}], {x, 0, nn}], x], 1] (* Geoffrey Critzer, Aug 01 2013 *)
weigh:= proc(p) proc(n) local x, k; coeff(series(mul((1+x^k)^p(k), k=1..n), x, n+1), x, n) end end: wsh:= p-> n-> weigh(p)(n-1): f:= n-> `if`(n>0 and n<12, [1$3, 2$5, 1$3][n], 0): a:= wsh(f)-f: seq(a(n), n=5..97); # Alois P. Heinz, Sep 10 2008
f[n_]:=Nest[CoefficientList[Series[Product[(1+x^i)^#[[i]],{i,1,Length[#]}],{x,0,50}],x]&,{1},n];Drop[f[4]-PadRight[f[3],Length[f[4]]],4] (* Geoffrey Critzer, Aug 01 2013 *)
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