A038348 Expansion of (1/(1-x^2))*Product_{m>=0} 1/(1-x^(2m+1)).

1, 1, 2, 3, 4, 6, 8, 11, 14, 19, 24, 31, 39, 49, 61, 76, 93, 114, 139, 168, 203, 244, 292, 348, 414, 490, 579, 682, 801, 938, 1097, 1278, 1487, 1726, 1999, 2311, 2667, 3071, 3531, 4053, 4644, 5313, 6070, 6923, 7886, 8971, 10190, 11561

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Number of partitions of n+2 with exactly one even part. - Vladeta Jovovic, Sep 10 2003

Also, number of partitions of n with at most one even part. - Vladeta Jovovic, Sep 10 2003

Also total number of parts, counted without multiplicity, in all partitions of n into odd parts, offset 1. - Vladeta Jovovic, Mar 27 2005

a(n) = Sum_{k>=1} k*A116674(n+1,k). - Emeric Deutsch, Feb 22 2006

Equals row sums of triangle A173305. - Gary W. Adamson, Feb 15 2010

Equals partial sums of A025147 (observed by Jonathan Vos Post, proved by several correspondents).

Conjecture: The n-th derivative of Gamma(x+1) at x = 0 has a(n+1) terms. For example, d^4/dx^4_(x = 0) Gamma(x+1) = 8*eulergamma*zeta(3) + eulergamma^4 + eulergamma^2*Pi^2 + 3*Pi^4/20 which has a(5) = 4 terms. - David Ulgenes, Dec 05 2023

Examples

			From _Gus Wiseman_, Sep 23 2019: (Start)
Also the number of integer partitions of n that are strict except possibly for any number of 1's. For example, the a(1) = 1 through a(7) = 11 partitions are:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (31)    (32)     (42)      (43)
             (111)  (211)   (41)     (51)      (52)
                    (1111)  (311)    (321)     (61)
                            (2111)   (411)     (421)
                            (11111)  (3111)    (511)
                                     (21111)   (3211)
                                     (111111)  (4111)
                                               (31111)
                                               (211111)
                                               (1111111)
(End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Cristina Ballantine and Mircea Merca, On identities of Watson type, Ars Mathematica Contemporanea (2019) Vol. 17, 277-290.
• Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
• P. Flajolet and B. Salvy, Euler sums and contour integral representations, Experimental Mathematics, Vol. 7 Issue 1 (1998).
• J. Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc. (N.S.), 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 2, with k=1. - _N. J. A. Sloane_, Aug 31 2014
• Rebekah Ann Gilbert, A Fine Rediscovery, 2014.
• Amrik Singh Nimbran and Paul Levrie, Series of the form Sum {a_n*binomial(2n, n)}, Math. Student (2023) Vol. 92, Nos. 3-4, 155-173. See pp. 10, 16.

Crossrefs

Cf. A067588, A090867, A116674, A173305.

Cf. A000009, A007360, A051424, A259936, A266591, A302569, A306200.

Programs

Maple

f:=1/(1-x^2)/product(1-x^(2*j-1),j=1..32): fser:=series(f,x=0,62): seq(coeff(fser,x,n),n=0..58); # Emeric Deutsch, Feb 22 2006

Mathematica

mmax = 47; CoefficientList[ Series[ (1/(1-x^2))*Product[1/(1-x^(2m+1)), {m, 0, mmax}], {x, 0, mmax}], x] (* Jean-François Alcover, Jun 21 2011 *)

SageMath

# uses[EulerTransform from A166861]
def g(n): return n % 2 if n > 2 else 1
a = EulerTransform(g)
print([a(n) for n in range(48)]) # Peter Luschny, Dec 04 2020

Formula

a(n) = A036469(n) - a(n-1) = Sum_{k=0..n} (-1)^k*A036469(n-k). - Vladeta Jovovic, Sep 10 2003

a(n) = A000009(n) + a(n-2). - Vladeta Jovovic, Feb 10 2004

G.f.: 1/((1-x^2)*Product_{j>=1} (1 - x^(2*j-1))). - Emeric Deutsch, Feb 22 2006

From Vaclav Kotesovec, Aug 16 2015: (Start)

a(n) ~ (1/2) * A036469(n).

a(n) ~ 3^(1/4) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)). (End)

Euler transform of the sequence [1, 1, period(1, 0)] (A266591). - Georg Fischer, Dec 04 2020