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A174065 Convolved with its aerated variant = A000041. 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Alois P. Heinz)

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 (k-1) interpolated zeros. Next term in left column = n-th term in the "target sequence" S(n) (in this case A000041) minus (sum of terms in n-th row for columns >1). Place the latter term in the heading filling in missing terms.

G.f.: Product_{i>=1, j>=0}  (1 + x^(i*4^j)). - Ilya Gutkovskiy, Sep 23 2019

a(n) ~ exp(2*Pi*sqrt(n)/3) / (2^(11/8) * 3^(3/4) * n^(7/8)). - Vaclav Kotesovec, Sep 24 2019

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.

MAPLE

p:= combinat[numbpart]:

a:= proc(n) option remember; `if`(n=0, 1, p(n)-add(a(j)*

      `if`(irem(n-j, 2, 'r')=1, 0, a(r)), j=0..n-1))

    end:

seq(a(n), n=0..61);  # Alois P. Heinz, Jul 27 2019

MATHEMATICA

nmax = 60; CoefficientList[Series[Product[QPochhammer[-1, x^(4^j)]/2, {j, 0, Log[nmax]/Log[4]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 24 2019 *)

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 from R. J. Mathar, Mar 18 2010

Offset corrected by Alois P. Heinz, Jul 27 2019

STATUS

approved

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Last modified December 1 12:50 EST 2020. Contains 338833 sequences. (Running on oeis4.)