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A038348 Expansion of (1/(1-x^2))*Product(1/(1-x^(2m+1)), m=0..infinity). 12
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*A116674(n+1,k),k>=1). - 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).

REFERENCES

Fulman, Jason. 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, http://www.math.uiuc.edu/~rgilber1/AFineRediscovery_Gilbert.pdf, 2014.

LINKS

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

P. Flajolet and B. Salvy, Euler sums and contour integral representations, Experimental Mathematics, Vol. 7 Issue 1 (1998)

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)

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.

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 23 09:38 EST 2017. Contains 295115 sequences.