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 A174068 Convolved with its aerated variant of two zeros between terms = A000041. 3
 1, 1, 2, 2, 4, 5, 7, 9, 13, 17, 23, 29, 38, 48, 62, 77, 98, 121, 153, 187, 233, 283, 349, 422, 515, 620, 751, 900, 1083, 1291, 1544, 1832, 2180, 2576, 3050, 3590, 4234, 4965, 5830, 6813, 7971, 9286, 10824, 12572, 14608, 16921, 19600, 22640, 26150, 30130, 34709 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Considered k=3 in an infinite set of convolution sequences: (aerated with one zero, A174065; two zeros, A174068); such that A000041 =   (1, 1, 2, 3, 5, 7, 11,...) = (1, 1, 2, 2, 4, 5, 7, 9, 13, 17,...) *   (1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 4, 0, 0, 5, 0, 0, 7, 0, 0,...). LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Alois P. Heinz) FORMULA Refer to A174065, and A174066, the case for k=3. The sequence = left border of a triangle generated from 3 rules: row sums = A000041; columns >1 are shifted down thrice from previous column; column terms are derived from self-convolution of left border, (with the left border placed at top as a heading). A(x)*A(x^3) = A000041(x) for the generating functions. - R. J. Mathar, Mar 18 2010 Expansion of f(-x^3)/f(-x) * f(-x^27)/f(-x^9) * f(-x^243)/f(-x^27) * ... where f(-x) is a Ramanujan theta function. - Michael Somos, Jun 07 2012 a(n) ~ exp(Pi*sqrt(n/2)) / (2^(19/8) * 3^(1/8) * n^(7/8)). - Vaclav Kotesovec, Sep 24 2019 EXAMPLE The triangle heading and first few rows of the triangle = 1, 1, 2, 2, 4, 5, 7,... 1; 1; 2; 2, 1; 4, 1; 5, 2; 7, 2, 2; ... G.f. = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 5*x^5 + 7*x^6 + 9*x^7 + 13*x^8 + 17*x^9 + ... MAPLE p:= combinat[numbpart]: a:= proc(n) option remember; `if`(n=0, 1, p(n)-add(a(j)*       `if`(irem(n-j, 3, 'r')>0, 0, a(r)), j=0..n-1))     end: seq(a(n), n=0..61);  # Alois P. Heinz, Jul 27 2019 MATHEMATICA a[n_] := a[n] = If[n == 0, 1, PartitionsP[n] - Sum[a[j]*If[Mod[n-j, 3] > 0, 0, a[(n-j)/3]], {j, 0, n-1}]]; a /@ Range[0, 61] (* Jean-François Alcover, May 17 2020, after Maple *) CROSSREFS Cf. A000041, A174065, A174066, A174067. Sequence in context: A128663 A206557 A240508 * A135833 A137200 A026930 Adjacent sequences:  A174065 A174066 A174067 * A174069 A174070 A174071 KEYWORD nonn AUTHOR Gary W. Adamson, Mar 06 2010 EXTENSIONS More terms from R. J. Mathar, Mar 18 2010 STATUS approved

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Last modified October 20 19:04 EDT 2020. Contains 337905 sequences. (Running on oeis4.)