A007690 Number of partitions of n in which no part occurs just once. (Formerly M0167)

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

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

N. J. A. Sloane, Robert G. Wilson v

Extensions

Minor edits by Vaclav Kotesovec, Aug 23 2015

Status

approved