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%I #18 Sep 23 2020 02:46:16
%S 1,2,4,8,17,36,78,170,375,833,1870,4229,9654,22223,51622,120961,
%T 286029,682398,1642821,3990231,9777678,24166327,60233185,151350709,
%U 383287499,977918150,2512805727,6500178867,16921248231,44310852884,116678914575
%N INVERT transform of A000055.
%C 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
%H Alois P. Heinz, <a href="/A157904/b157904.txt">Table of n, a(n) for n = 0..700</a>
%F INVERT transform of A000055: (1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106,...).
%e 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.
%p 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
%t 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 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];
%t a[n_] := a[n] = If[n <= 0, 1, Sum[t[i] a[n - i - 1], {i, 0, n}]];
%t a /@ Range[0, 30] (* _Jean-François Alcover_, Sep 22 2020, after _Alois P. Heinz_ *)
%Y Cf. A000055, A157905.
%K nonn
%O 0,2
%A _Gary W. Adamson_, Mar 08 2009
%E More terms from _Alois P. Heinz_, Mar 31 2009
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