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A038348 Expansion of (1/(1-x^2))*Product_{m>=0} 1/(1-x^(2m+1)). 17
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cristina Ballantine, Mircea Merca, On identities of Watson type, Ars Mathematica Contemporanea (2019) Vol. 17, 277-290.

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.

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)

EXAMPLE

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)

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-Fran├žois Alcover, Jun 21 2011 *)

CROSSREFS

Cf. A067588, A090867, A116674, A173305.

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

Sequence in context: A261154 A233693 A003412 * A239467 A035945 A094707

Adjacent sequences:  A038345 A038346 A038347 * A038349 A038350 A038351

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 24 12:22 EST 2020. Contains 338612 sequences. (Running on oeis4.)