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A035363 Number of partitions of n into even parts. 18
1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 7, 0, 11, 0, 15, 0, 22, 0, 30, 0, 42, 0, 56, 0, 77, 0, 101, 0, 135, 0, 176, 0, 231, 0, 297, 0, 385, 0, 490, 0, 627, 0, 792, 0, 1002, 0, 1255, 0, 1575, 0, 1958, 0, 2436, 0, 3010, 0, 3718, 0, 4565, 0, 5604, 0, 6842, 0, 8349, 0, 10143, 0, 12310, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Convolved with A036469 = A000070. - Gary W. Adamson, Jun 09 2009

Note that these partitions are located in the head of the last section of the set of partitions of n (see A135010). - Omar E. Pol, Nov 20 2009

Number of symmetric unimodal compositions of n+2 where the maximal part appears twice, see example. Also number of symmetric unimodal compositions of n where the maximal part appears an even number of times. - Joerg Arndt, Jun 11 2013

Number of partitions of n having parts of even multiplicity. These are the conjugates of the partitions from the definition. Example: a(8)=5 because we have [4,4],[3,3,1,1],[2,2,2,2],[2,2,1,1,1,1], and [1,1,1,1,1,1,1,1]. - Emeric Deutsch, Jan 27 2016

REFERENCES

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education, Vol. 31, No. 1, pp. 24-28, Winter 1997.  MathEduc Database (Zentralblatt MATH, 1997c.01891).

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II,   Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17.  Zentralblatt MATH, Zbl 1071.05501.

LINKS

Table of n, a(n) for n=0..69.

FORMULA

G.f.: prod(1/(1-x^k), k even)

Convolution with the number of partitions into distinct parts (A000009, which is also number of partitions into odd parts) gives the number of partitions (A000041). - Franklin T. Adams-Watters, Jan 06 2006

If n is even then a(n)=A000041(n/2) otherwise a(n)=0. - Omar E. Pol, Nov 20 2009

G.f.: 1 + x^2*(1 - G(0))/(1-x^2) where G(k) =  1 - 1/(1-x^(2*k+2))/(1-x^2/(x^2-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 23 2013

a(n) = A096441(n) - A000009(n), n >= 1. - Omar E. Pol, Aug 16 2013

EXAMPLE

From Joerg Arndt, Jun 11 2013: (Start)

There are a(12)=11 symmetric unimodal compositions of 12+2=14 where the maximal part appears twice:

01:  [ 1 1 1 1 1 2 2 1 1 1 1 1 ]

02:  [ 1 1 1 1 3 3 1 1 1 1 ]

03:  [ 1 1 1 4 4 1 1 1 ]

04:  [ 1 1 2 3 3 2 1 1 ]

05:  [ 1 1 5 5 1 1 ]

06:  [ 1 2 4 4 2 1 ]

07:  [ 1 6 6 1 ]

08:  [ 2 2 3 3 2 2 ]

09:  [ 2 5 5 2 ]

10:  [ 3 4 4 3 ]

11:  [ 7 7 ]

There are a(14)=15 symmetric unimodal compositions of 14 where the maximal part appears an even number of times:

01:  [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]

02:  [ 1 1 1 1 1 2 2 1 1 1 1 1 ]

03:  [ 1 1 1 1 3 3 1 1 1 1 ]

04:  [ 1 1 1 2 2 2 2 1 1 1 ]

05:  [ 1 1 1 4 4 1 1 1 ]

06:  [ 1 1 2 3 3 2 1 1 ]

07:  [ 1 1 5 5 1 1 ]

08:  [ 1 2 2 2 2 2 2 1 ]

09:  [ 1 2 4 4 2 1 ]

10:  [ 1 3 3 3 3 1 ]

11:  [ 1 6 6 1 ]

12:  [ 2 2 3 3 2 2 ]

13:  [ 2 5 5 2 ]

14:  [ 3 4 4 3 ]

15:  [ 7 7 ]

(End)

a(8)=5 because we  have [8], [6,2], [4,4], [4,2,2], and [2,2,2,2]. - Emeric Deutsch, Jan 27 2016

MAPLE

ZL:= [S, {C = Cycle(B), S = Set(C), E = Set(B), B = Prod(Z, Z)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=0..69); # Zerinvary Lajos, Mar 26 2008

g := 1/mul(1-x^(2*k), k = 1 .. 100): gser := series(g, x = 0, 80): seq(coeff(gser, x, n), n = 0 .. 78); # Emeric Deutsch, Jan 27 2016

CROSSREFS

Bisection (even part) gives the partition numbers A000041.

Column k=0 of A103919, A264398.

Cf. A036469, A000070.

Cf. A135010, A138121.

Sequence in context: A240144 A240145 A240146 * A241645 A266774 A079977

Adjacent sequences:  A035360 A035361 A035362 * A035364 A035365 A035366

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard

STATUS

approved

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Last modified May 24 21:25 EDT 2017. Contains 287008 sequences.