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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

Table of n, a(n) for n=1..46.

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[0].

# 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.)