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A006171 Number of factorization patterns of polynomials of degree n over integers.
(Formerly M2479)
79
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, 6924303 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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). - Franklin T. Adams-Watters, Jun 19 2006
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, ...). - Gary W. Adamson, Jun 16 2009
Also equals the number of distinct (up to unitary similarity) unital *-subalgebras of the n X n complex matrices. A unital *-subalgebra is a subspace that is closed under multiplication and the conjugate transpose, and which contains the identity matrix (see A215905 and A215925). - Nathaniel Johnston, Aug 27 2012
Also equals the number of partitions having parts consisting of runs of equal parts. - Gregory L. Simay, May 25 2017
Also equals the number of generalized partitions of n when there are d(a) different types of a, (a = 1,2,3,...), where d(n) is the number of divisors of n. a(3)=5 because there are 5 partitions of 3 with "d(a) copies of a", namely (3_1), (3_2), (2_1, 1_1), (2_2, 1_1), (1_1, 1_1, 1_1). - Augustine O. Munagi, Jun 13 2022
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).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
A. K. Agarwal and G. L. Mullen, Partitions with "d(a) copies of a", J. Combin. Theory Ser. A, 48(1)(1988), 120-135.
N. A. Brigham, A General Asymptotic Formula for Partition Functions, Proc. Amer. Math. Soc., vol. 1 (1950), pp. 182-191.
N. J. A. Sloane, Transforms
FORMULA
From Vladeta Jovovic, Apr 21 2001: (Start)
Euler transform of tau(n), tau(n) = the number of divisors of n, cf. A000005.
G.f.: Product_{k>=1} (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. (End)
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. - Franklin T. Adams-Watters, Jun 19 2006
G.f.: A(x) = exp( Sum_{n>=1} sigma(n)*x^n/(1-x^n)/n ). - Paul D. Hanna, Mar 28 2009
From Paul D. Hanna, Oct 19 2011: (Start)
Logarithmic derivative yields A060640.
G.f.: A(x) = exp( Sum_{n>=1} A060640(n)*x^n/n ), where A060640(n) = Sum_{d|n} d*sigma(n/d). (End)
G.f.: 1/Product_{n>=1} E(q^n) where E(q) = Product_{n>=1} (1-q^n). - Joerg Arndt, Feb 27 2014
log(a(n)) ~ Pi * sqrt(n*log(n)/3) [Brigham, 1950]. - Vaclav Kotesovec, Jan 04 2017
a(n) ~ exp(Pi*sqrt(n/(3*log(n))) * (log(n) - log(log(n))/2 + gamma + 6*Zeta'(2)/Pi^2 + log(2/Pi) + log(3)/2)) * Pi^(1/4) * (log(n))^(1/8) / (2^(3/4) * 3^(1/8) * n^(5/8)), where gamma is the Euler-Mascheroni constant (A001620) and Zeta'(2) = -0.9375482543158437537... (see A073002) [user Lucia, MathOverflow, 2014]. - Vaclav Kotesovec, Jan 05 2017
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.
For n=4 we have the following 11 partitions, with the additive runs indicated by "[]": [4], [3]+[1], [2+2], [2]+[2], [2]+[1+1], [2]+[1]+[1], [1+1+1+1], [1+1+1]+[1], [1+1]+[1+1], [1+1]+[1]+[1], [1]+[1]+[1]+[1]. - Gregory L. Simay, May 25 2017
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..40); # Alois P. Heinz, Sep 08 2008
MATHEMATICA
max = 50; gf[x_] := Product[(1 - x^k)^-DivisorSigma[0, k], {k, 1, max}]; CoefficientList[ Series[gf[x], {x, 0, max}], x] (* Jean-François Alcover, Nov 23 2011 *)
nmax = 50; s = 1 - x; Do[s *= Sum[Binomial[DivisorSigma[0, k], j]*(-1)^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]]; , {k, 2, nmax}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 28 2018, the fastest *)
nmax = 50; CoefficientList[Series[Product[Sum[PartitionsP[k]*x^(j*k), {k, 0, nmax/j}], {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 26 2020 *)
PROG
(PARI) {a(n) = if(n<0, 0, polcoeff( 1 / prod(k=1, n, (1 - x^k + x * O(x^n))^numdiv(k)), n))}; /* Michael Somos, Apr 01 2003 */
(PARI) N=66; x='x+O('x^N); gf=1/prod(j=1, N, eta(x^j)); Vec(gf) \\ 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))} /* Paul D. Hanna, Mar 28 2009 */
(PARI) {A060640(n)=sumdiv(n, d, d*sigma(n/d))}
{a(n)=polcoeff(exp(sum(m=1, n+1, A060640(m)*x^m/m)+x*O(x^n)), n)} /* Paul D. Hanna, Oct 19 2011 */
CROSSREFS
Cf. A006167-A006170, A006171 (log).
Cf. A000219.
Sequence in context: A327044 A006170 A147071 * A261674 A319632 A060647
KEYWORD
nonn,nice
AUTHOR
STATUS
approved

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Last modified February 21 02:59 EST 2024. Contains 370219 sequences. (Running on oeis4.)