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A006171
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Number of factorization patterns of polynomials of degree n over integers.
(Formerly M2479)
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13
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1, 1, 3, 5, 11, 17, 34, 52, 94, 145, 244, 370, 603, 899, 1410, 2087, 3186, 4650, 6959, 10040, 14750, 21077, 30479, 43120, 61574, 86308, 121785, 169336, 236475, 326201, 451402, 618135, 848209, 1153733, 1571063, 2123325, 2871419, 3857569, 5182999
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Number of partitions of n where there are unlimited distinguishable but unlabeled objects of each size. E.g., in splitting 2 into two parts of size 1, we distinguish whether the same object is used for each part. Also number of factorization patterns over rationals, or many other UFDs (but not over real or complex numbers). - Frank Adams-Watters , Jun 19 2006
Contribution from Gary W. Adamson, Jun 16 2009: (Start)
Equals the "aerate and convolve" convergent of A000041 =
(1, 1, 2, 3, 5, 7, 11,...) * (1, 0, 1, 0, 2, 0, 3, 0, 5,...) * (1, 0, 0, 1, 0, 0, 2, 0, 0, 3,...) (End)
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REFERENCES
| R. A. Hultquist, G. L. Mullen and H. Niederreiter, Association schemes and derived PBIB designs of prime power order, Ars. Combin., 25 (1988), 65-82.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
N. J. A. Sloane, Transforms
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FORMULA
| Euler transform of tau(n), tau(n)=the number of divisors of n, cf. A000005. G.f.: Product_{k=1..infinity} (1 - x^k)^(-tau(k)). a(n)=1/n*Sum_{k=1..n} a(n-k)*b(k), n>1, a(0)=1, b(k)=Sum_{d|k} d*tau(d), cf. A060640. - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 21 2001
a(n) = sum_{<b(i)^k(i)> partition of n} product p(k(i)), where p(n) is the partition function A000041. E.g., for the partition [4,2^3,1^4], the product is p(1)*p(3)*p(4) = 1*3*5 = 15. - Frank Adams-Watters , Jun 19 2006
G.f.: A(x) = exp( Sum_{n>=1} sigma(n)*x^n/(1-x^n)/n ). [From Paul D. Hanna, Mar 28 2009]
Contribution from Paul D. Hanna, Oct 19 2011: (Start)
Logarithmic derivative yields A006171.
G.f.: A(x) = exp( Sum_{n>=1} A006171(n)*x^n/n ), where A006171(n) = Sum_{d|n} d*sigma(n/d). (End)
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EXAMPLE
| For n=3 we have 3=(3*1)=(1*3)=(2*1)+(1*1)=(1*2)+(1*1)=(1*1)+(1*1)+(1*1) so a(3)=5.
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MAPLE
| with (numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add (add (d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:=etr (tau): seq (a(n), n=0..38); [From Alois P. Heinz, Sep 08 2008]
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MATHEMATICA
| max = 38; gf[x_] := Product[(1 - x^k)^-DivisorSigma[0, k], {k, 1, max}]; CoefficientList[ Series[gf[x], {x, 0, max}], x](* From Jean-François Alcover, Nov 23 2011 *)
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PROG
| (PARI) a(n)=if(n<0, 0, polcoeff(1/prod(k=1, n, (1-x^k+x*O(x^n))^numdiv(k)), n))
(PARI) N=66; x='x+O('x^N); /* that many terms */
gf=1/prod(j=1, N, eta(x^j));
Vec(gf) /*show terms */ /* Joerg Arndt, May 03 2008 */
(PARI) {a(n)=if(n==0, 1, polcoeff(exp(sum(m=1, n, sigma(m)*x^m/(1-x^m+x*O(x^n))/m)), n))} /* From Paul D. Hanna, Mar 28 2009 */
(PARI) {A006171(n)=sumdiv(n, d, d*sigma(n/d))}
{a(n)=polcoeff(exp(sum(m=1, n+1, A006171(m)*x^m/m)+x*O(x^n)), n)} /* From Paul D. Hanna, Oct 19 2011 */
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CROSSREFS
| Cf. A000005, A060640, A061255, A061256, A001970, A061257.
Cf. A006167-A006170, A006171 (log).
Cf. A000041, A115621.
Sequence in context: A091610 A006170 A147071 * A060647 A155989 A125557
Adjacent sequences: A006168 A006169 A006170 * A006172 A006173 A006174
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KEYWORD
| nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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