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 A147541 Result of using the primes as coefficients in an infinite polynomial series in x and then expressing this series as (1+x)(1+a(1)*x)(1+a(2)*x^2) ... . 9
 1, 2, 1, 3, 2, -4, 2, 5, 4, -6, 4, 4, 10, -36, 18, 45, 34, -72, 64, -24, 124, -358, 258, 170, 458, -1260, 916, 148, 1888, -4296, 3690, 887, 7272, -17616, 14718, -5096, 29610, -67164, 58722, -26036, 119602, -244496, 242256, -104754, 487352, -1029384 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS This is the PPE (power product expansion) of A036467. - R. J. Mathar, Feb 01 2010 LINKS H. Gingold, A note on reduction of operations via power product approximations, Utilitas Math. 37 (1990), 79-89. [From R. J. Mathar, Nov 10 2008] H. Gingold and A. Knopfmacher, Analytic properties of power product expansions, Canad. J. Math. 47 (1995), 1219-1239. [From R. J. Mathar, Nov 10 2008] H. Gingold, A. Knopfmacher and D. Lubinsky, The zero distribution of the partial products of power product expansions, Analysis 13 (1993), 133-157. [From R. J. Mathar, Nov 10 2008] EXAMPLE From the primes, construct the series 1+2x+3x^2+5x^3+7x^4+... Divide this by (1+x) to get the quotient (1+a(1)x+...), which here gives a(1)=1. Then divide this quotient by (1+a(1)x), i.e. here (1+x), to get (1+a(2)x^2+...), giving a(2)=2. MAPLE From R. J. Mathar, Feb 01 2010: (Start) # Partition n into a set of distinct positive integers, the maximum one # being m. # Example: partitionsQ(7, 5) returns [[2, 5], [3, 4], [1, 2, 4]] ; # Richard J. Mathar, 2008-11-10 partitionsQ := proc(n, m) local p, t, rec, q; p := [] ; # take 't' of the n and recursively determine the partitions of # what has been left over. for t from min(m, n) to 1 by -1 do # Since we are only considering partitions into distinct parts, # the triangular numbers set a lower bound on the t. if t*(t+1)/2 >= n then rec := partitionsQ(n-t, t-1) ; if nops(rec) = 0 then p := [op(p), [t]] ; else for q in rec do p := [op(p), [op(q), t]] ; end do: end if; end if; end do: RETURN(p) ; end proc: # Power product expansion of L. # L is a list starting with 1, which is considered L. # Returns the list [a(1), a(2), ..] such that # product_(i=1, 2, ..) (1+a(i)x^i) = sum_(j=0, 1, 2, ...) L[j]x^j. # Richard J. Mathar, 2008-11-10 ppe := proc(L) local pro, i, par, swithi, snoti, m, p, k ; pro := [] ; for i from 1 to nops(L)-1 do par := partitionsQ(i, i) ; swithi := 0 ; snoti := 0 ; for p in par do if i in p then m := 1 ; for k from 1 to nops(p)-1 do m := m*op(op(k, p), pro) ; end do; swithi := swithi+m ; else snoti := snoti+mul( op(k, pro), k=p) ; end if; end do: pro := [op(pro), (op(i+1, L)-snoti)/swithi] ; end do: RETURN(pro) ; end proc: read("transforms") ; A147541 := proc(nmax) local L, L1, L2 ; L := [1, seq(ithprime(n), n=1..nmax)] ; L1 := [seq((-1)^n, n=0..nmax+10)] ; A036467 := CONV(L, L1) ; ppe(A036467) ; end: A147541(47) ; (End) CROSSREFS Cf. A000040, A147542. Sequence in context: A106466 A130722 A308307 * A308308 A024162 A334677 Adjacent sequences:  A147538 A147539 A147540 * A147542 A147543 A147544 KEYWORD sign AUTHOR Neil Fernandez, Nov 06 2008 EXTENSIONS Extended by R. J. Mathar, Feb 01 2010 STATUS approved

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Last modified August 6 18:52 EDT 2020. Contains 336256 sequences. (Running on oeis4.)