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

0,2

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

Cf. A000055, A157905.

Sequence in context: A190162 A275691 A251691 * A182901 A002845 A072925

Adjacent sequences: A157901 A157902 A157903 * A157905 A157906 A157907

Keyword

nonn

Author

Gary W. Adamson, Mar 08 2009

Extensions

More terms from Alois P. Heinz, Mar 31 2009

Status

approved