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 A001452 Number of 5-line partitions of n. (Formerly M2564 N1015) 11
 1, 1, 3, 6, 13, 24, 47, 83, 152, 263, 457, 768, 1292, 2118, 3462, 5564, 8888, 14016, 21973, 34081, 52552, 80331, 122078, 184161, 276303, 411870, 610818, 900721, 1321848, 1929981, 2805338, 4058812, 5847966, 8390097, 11990531, 17069145, 24210571, 34215537, 48190451, 67644522 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Planar partitions into at most five rows. - Joerg Arndt, May 01 2013 Number of partitions of n where there are k sorts of parts k for k<=4 and 5 sorts all other parts. - Joerg Arndt, Mar 15 2014 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (first 1000 terms from Alois P. Heinz) M. S. Cheema and B. Gordon, Some remarks on two- and three-line partitions, Duke Math. J., 31 (1964), 267-273. Vaclav Kotesovec, Graph - The asymptotic ratio (50000 terms) P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116. FORMULA G.f.: 1 / prod(k>=1, (1-x^k)^min(k,5) ). - Sean A. Irvine, Jul 24 2012 a(n) ~ 15625 * Pi^10 * sqrt(5) * exp(Pi*sqrt(10*n/3)) / (2592 * sqrt(3) * n^7). - Vaclav Kotesovec, Oct 28 2015 MAPLE with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(add(       min(d, 5)*d, d=divisors(j))*a(n-j), j=1..n)/n)     end: seq(a(n), n=0..45);  # Alois P. Heinz, Mar 15 2014 MATHEMATICA a[n_] := a[n] = If[n == 0, 1, Sum[Sum[Min[d, 5]*d, {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *) nmax = 40; CoefficientList[Series[(1-x)^4 * (1-x^2)^3 * (1-x^3)^2 * (1-x^4) * Product[1/(1-x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 28 2015 *) PROG (PARI) x='x+O('x^66); r=5; Vec( prod(k=1, r-1, (1-x^k)^(r-k)) / eta(x)^r ) \\ Joerg Arndt, May 01 2013 (MAGMA) m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)^4*(1-x^2)^3*(1-x^3)^2*(1-x^4)/(&*[1-x^j: j in [1..2*m]])^5 )); // G. C. Greubel, Dec 06 2018 (Sage) s=((1-x)^4*(1-x^2)^3*(1-x^3)^2*(1-x^4)/prod(1-x^j for j in (1..60))^5).series(x, 50); s.coefficients(x, sparse=False) # G. C. Greubel, Dec 06 2018 CROSSREFS A row of the array in A242641. Sequences "number of r-line partitions": A000041 (r=1), A000990 (r=2), A000991 (r=3), A002799 (r=4), A001452 (r=5), A225196 (r=6), A225197 (r=7), A225198 (r=8), A225199 (r=9). Sequence in context: A293076 A293421 A018081 * A005405 A225196 A301597 Adjacent sequences:  A001449 A001450 A001451 * A001453 A001454 A001455 KEYWORD nonn AUTHOR EXTENSIONS More terms from Sean A. Irvine, Jul 24 2012 Prepended a(0)=1, Joerg Arndt, May 01 2013 STATUS approved

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Last modified April 24 19:49 EDT 2019. Contains 322446 sequences. (Running on oeis4.)