A157904 INVERT transform of A000055.

1, 2, 4, 8, 17, 36, 78, 170, 375, 833, 1870, 4229, 9654, 22223, 51622, 120961, 286029, 682398, 1642821, 3990231, 9777678, 24166327, 60233185, 151350709, 383287499, 977918150, 2512805727, 6500178867, 16921248231, 44310852884, 116678914575

Offset: 0

Author

Gary W. Adamson, Mar 08 2009

Keywords

• nonn

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

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..700

Crossrefs

Cf. A000055, A157905.

Programs

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 *)

Formula

INVERT transform of A000055: (1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106,...).

Extensions

More terms from Alois P. Heinz, Mar 31 2009