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A157904 INVERT transform of A000055. 3
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified May 14 07:13 EDT 2021. Contains 343879 sequences. (Running on oeis4.)