OFFSET
0,2
COMMENTS
Note that the correct INVERT transform of A000055 (recognizing the offsets) would be 1, 1, 2, 4, 9, 20, 46, 106, 248, 583, 1386,... - R. J. Mathar, Sep 20 2020
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..700
FORMULA
INVERT transform of A000055: (1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106,...).
EXAMPLE
a(3) = 8 = (1, 1, 1) dot (1, 2, 4) + 1 = 7 + 1 = 8; where the operation uses ascending terms of A000055: (1, 1, 1, 1, 2, 3, 6, 11,...) and an equal number of ongoing descending terms of A157904. Take the dot product and add to the next term of A000055. a(4) = 17 = (1, 1, 1, 1) dot (1, 2, 4, 8) + 2 = 15 + 2.
MAPLE
with(numtheory): b:= proc(n) option remember; local d, j; if n<=1 then n else (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/ (n-1) fi end: t:= proc(n) option remember; local k; `if`(n=0, 1, b(n)- (add(b(k) *b(n-k), k=1..n-1) -`if`(type(n, odd), 0, b(n/2)))/2) end: a:= proc(n) option remember; local i; if n<=0 then 1 else add(t(i)*a(n-i-1), i=0..n) fi end: seq(a(n), n=0..35); # Alois P. Heinz, Mar 31 2009
MATHEMATICA
b[n_] := b[n] = If[n <= 1, n, Sum[Sum[d b[d], {d, Divisors[j]}] b[n - j], {j, 1, n - 1}]/(n - 1)];
t[n_] := t[n] = If[n == 0, 1, b[n] - (Sum[b[k] b[n - k], {k, 1, n - 1}] - If[OddQ[n], 0, b[n/2]])/2];
a[n_] := a[n] = If[n <= 0, 1, Sum[t[i] a[n - i - 1], {i, 0, n}]];
a /@ Range[0, 30] (* Jean-François Alcover, Sep 22 2020, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gary W. Adamson, Mar 08 2009
EXTENSIONS
More terms from Alois P. Heinz, Mar 31 2009
STATUS
approved