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 A007690 Number of partitions of n in which no part occurs just once. (Formerly M0167) 60
 1, 0, 1, 1, 2, 1, 4, 2, 6, 5, 9, 7, 16, 11, 22, 20, 33, 28, 51, 42, 71, 66, 100, 92, 147, 131, 199, 193, 275, 263, 385, 364, 516, 511, 694, 686, 946, 925, 1246, 1260, 1650, 1663, 2194, 2202, 2857, 2928, 3721, 3813, 4866, 4967, 6257, 6487, 8051, 8342, 10369 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Also number of partitions of n into parts, each larger than 1, such that consecutive integers do not both appear as parts. Example: a(6)=4 because we have [6], [4,2], [3,3] and [2,2,2]. - Emeric Deutsch, Feb 16 2006 Also number of partitions of n into parts divisible by 2 or 3. - Alexander E. Holroyd (holroyd(AT)math.ubc.ca), May 28 2008 Infinite convolution product of [1,0,1,1,1,1,1] aerated n-1 times. i.e. [1,0,1,1,1,1,1] * [1,0,0,0,1,0,1] * [1,0,0,0,0,0,1] * ... . - Mats Granvik, Aug 07 2009 REFERENCES G. E. Andrews, Number Theory, Dover Publications, 1994. page 197. MR1298627 G. E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976, p. 14, Example 9. I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.5.6). R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 242. P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 54, Article 300. 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 A. E. Holroyd, T. M. Liggett and D. Romik, Integrals, partitions and cellular automata, arXiv:math/0302216 [math.PR], 2003. Eric Weisstein's World of Mathematics, Partition Function P FORMULA G.f.: Product_{k>0 is a multiple of 2 or 3} (1/(1-x^k)). - Christian G. Bower, Jun 23 2000 G.f.: Product_{j>=1} (1+x^(3*j)) / (1-x^(2*j)). - Jon Perry, Mar 29 2004 Euler transform of period 6 sequence [0, 1, 1, 1, 0, 1, ...]. - Michael Somos, Apr 21 2004 G.f. is a period 1 Fourier series which satisfies f(-1 / (864 t)) = 1/6 (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A137566. - Michael Somos, Jan 26 2008 From Alois P. Heinz, Oct 09 2011: (Start) a(n) = A000041(n) - A183558(n). a(n) = A183568(n,0) - A183568(n,1). G.f.: Product_{j>0} (1-x^j+x^(2*j)) / (1-x^j). (End) a(n) ~ exp(2*Pi*sqrt(n)/3)/(6*sqrt(2)*n). - Vaclav Kotesovec, Sep 23 2015 a(n) = A000009(n/3) - Sum_{k>=1} (-1)^k a(n - k*(3*k +/- 1)). - Peter J. Taylor, May 16 2019 EXAMPLE a(6) = 4 because we have [3,3], [2,2,2], [2,2,1,1] and [1,1,1,1,1,1]. G.f. = 1 + x^2 + x^3 + 2*x^4 + x^5 + 4*x^6 + 2*x^7 + 6*x^8 + 5*x^9 + 9*x^10 + ... G.f. = q + q^49 + q^73 + 2*q^97 + q^121 + 4*q^145 + 2*q^169 + 6*q^193 + ... MAPLE G:= mul((1-x^j+x^(2*j))/(1-x^j), j=1..70): Gser:=series(G, x, 60): seq(coeff(Gser, x, n), n=0..54); # Emeric Deutsch, Feb 10 2006 MATHEMATICA nn=40; CoefficientList[Series[Product[1/(1-x^i)-x^i, {i, 1, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, Dec 02 2012 *) a[ n_] := SeriesCoefficient[ QPochhammer[ x^6] / (QPochhammer[ x^2] QPochhammer[ x^3]), {x, 0, n}]; (* Michael Somos, Feb 22 2015 *) nmax = 60; CoefficientList[Series[Product[(1 + x^(3*k))/(1 - x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 23 2015 *) Table[Length@Select[Tally /@ IntegerPartitions@n, AllTrue[#, Last[#] > 1 &] &], {n, 0, 54}] (* Robert Price, Aug 17 2020 *) PROG (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A) / (eta(x^2 + A) * eta(x^3 + A)), n))}; /* Michael Somos, Apr 21 2004 */ CROSSREFS Cf. A000041, A055922, A055923, A114917, A114918, A183558, A183568. Cf. A100405, A160974-A160990. Sequence in context: A161268 A176837 A339241 * A239960 A330001 A292402 Adjacent sequences: A007687 A007688 A007689 * A007691 A007692 A007693 KEYWORD nonn AUTHOR EXTENSIONS Minor edits by Vaclav Kotesovec, Aug 23 2015 STATUS approved

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Last modified December 5 02:30 EST 2022. Contains 358572 sequences. (Running on oeis4.)