

A174065


Convolved with its aerated variant = A000041


8



1, 1, 1, 2, 3, 4, 5, 7, 9, 12, 15, 19, 25, 31, 38, 48, 60, 73, 89, 109, 133, 161, 193, 232, 279, 333, 395, 470, 558, 658, 775, 912, 1071, 1254, 1464, 1708, 1991, 2313, 2681, 3107, 3595, 4149, 4782, 5506, 6331, 7268, 8330, 9538, 10912, 12462, 14213, 16199
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OFFSET

1,4


COMMENTS

A000041 = (1, 1, 2, 3, 5, 7,...) = (1, 1, 1, 2, 3, 4, 5, 7,...) * (1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 7, 0, 9, 0,...).
The sequence diverges from A100853 after 16 terms; and is a conjectured Euler transform of A035263: (1, 0, 1, 1, 1, 0, 1, 0, 1,...).


LINKS

Table of n, a(n) for n=1..52.


FORMULA

Aerate and convolve sequences are generated by triangles (in this case A174066) in which ongoing terms are placed in the left column and at the top as a heading. Columns >1 are shifted down k times (k=2) in this case corresponding to (k1) interpolated zeros. Next term in left column = nth term in the "target sequence" S(n) (in this case A000041) minus (sum of terms in nth row for columns >1). Place the latter term in the heading filling in missing terms.


EXAMPLE

Heading at top, with triangle A174066 underneath (the generator for A174065):
1, 1, 1, 2, 3, 4,.... = heading
1;................... = 1
1;................... = 1
1, 1;................ = 2
2, 1;................ = 3
3, 1, 1;............. = 5
4, 2, 1;............. = 7
5, 3, 1, 2;.......... = 11
7, 4, 2, 2;.......... = 15
9, 5, 3, 2, 3;....... = 22
...
... where terms in the left column are the result of the two rules: multiply heading * left column, and row sums = partition numbers.
Thus leftmost term in column 8 must be 7 = 15  (4 + 2 + 2). Then the 7 is placed in its spot in the left column and as the next heading term.


CROSSREFS

Cf. A000041, A174066, A174067.
Sequence in context: A280909 A003413 A100853 * A121659 A096814 A039861
Adjacent sequences: A174062 A174063 A174064 * A174066 A174067 A174068


KEYWORD

nonn


AUTHOR

Gary W. Adamson, Mar 06 2010


EXTENSIONS

More terms. R. J. Mathar, Mar 18 2010


STATUS

approved



