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Convolved with its aerated variant = A000041.
12

%I #32 May 31 2024 14:35:07

%S 1,1,1,2,3,4,5,7,9,12,15,19,25,31,38,48,60,73,89,109,133,161,193,232,

%T 279,333,395,470,558,658,775,912,1071,1254,1464,1708,1991,2313,2681,

%U 3107,3595,4149,4782,5506,6331,7268,8330,9538,10912,12462,14213,16199

%N Convolved with its aerated variant = A000041.

%C 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, ...).

%C 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, ...).

%H Vaclav Kotesovec, <a href="/A174065/b174065.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..5000 from Alois P. Heinz)

%F 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.

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

%F a(n) ~ exp(2*Pi*sqrt(n)/3) / (2^(11/8) * 3^(3/4) * n^(7/8)). - _Vaclav Kotesovec_, Sep 24 2019

%F From _Seiichi Manyama_, May 31 2024: (Start)

%F G.f.: Product_{k>=1} (1 + x^k)^(valuation(k,4) + 1).

%F Let A(x) be the g.f. of this sequence, and B(x) be the g.f. of A000009, then B(x) = A(x)/A(x^4). (End)

%e Heading at top, with triangle A174066 underneath (the generator for A174065):

%e 1, 1, 1, 2, 3, 4,.... = heading

%e 1;................... = 1

%e 1;................... = 1

%e 1, 1;................ = 2

%e 2, 1;................ = 3

%e 3, 1, 1;............. = 5

%e 4, 2, 1;............. = 7

%e 5, 3, 1, 2;.......... = 11

%e 7, 4, 2, 2;.......... = 15

%e 9, 5, 3, 2, 3;....... = 22

%e ...

%e ... where terms in the left column are the result of the two rules: multiply heading * left column, and row sums = partition numbers.

%e 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.

%p p:= combinat[numbpart]:

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

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

%p end:

%p seq(a(n), n=0..61); # _Alois P. Heinz_, Jul 27 2019

%t 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 *)

%Y Cf. A000041, A327726, A373219, A373220, A373221.

%Y Cf. A000009, A115362, A174066, A174067.

%K nonn

%O 0,4

%A _Gary W. Adamson_, Mar 06 2010

%E More terms from _R. J. Mathar_, Mar 18 2010

%E Offset corrected by _Alois P. Heinz_, Jul 27 2019