This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A038348 #71 Feb 21 2024 10:34:13
%S A038348 1,1,2,3,4,6,8,11,14,19,24,31,39,49,61,76,93,114,139,168,203,244,292,
%T A038348 348,414,490,579,682,801,938,1097,1278,1487,1726,1999,2311,2667,3071,
%U A038348 3531,4053,4644,5313,6070,6923,7886,8971,10190,11561
%N A038348 Expansion of (1/(1-x^2))*Product_{m>=0} 1/(1-x^(2m+1)).
%C A038348 Number of partitions of n+2 with exactly one even part. - _Vladeta Jovovic_, Sep 10 2003
%C A038348 Also, number of partitions of n with at most one even part. - _Vladeta Jovovic_, Sep 10 2003
%C A038348 Also total number of parts, counted without multiplicity, in all partitions of n into odd parts, offset 1. - _Vladeta Jovovic_, Mar 27 2005
%C A038348 a(n) = Sum_{k>=1} k*A116674(n+1,k). - _Emeric Deutsch_, Feb 22 2006
%C A038348 Equals row sums of triangle A173305. - _Gary W. Adamson_, Feb 15 2010
%C A038348 Equals partial sums of A025147 (observed by _Jonathan Vos Post_, proved by several correspondents).
%C A038348 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
%H A038348 Alois P. Heinz, <a href="/A038348/b038348.txt">Table of n, a(n) for n = 0..1000</a>
%H A038348 Cristina Ballantine and Mircea Merca, <a href="https://doi.org/10.26493/1855-3974.1782.127">On identities of Watson type</a>, Ars Mathematica Contemporanea (2019) Vol. 17, 277-290.
%H A038348 Kevin Beanland and Hung Viet Chu, <a href="https://arxiv.org/abs/2311.01926">On Schreier-type Sets, Partitions, and Compositions</a>, arXiv:2311.01926 [math.CO], 2023.
%H A038348 P. Flajolet and B. Salvy, <a href="http://www.emis.de/journals/EM/expmath/volumes/7/7.html">Euler sums and contour integral representations</a>, Experimental Mathematics, Vol. 7 Issue 1 (1998).
%H A038348 J. Fulman, <a href="http://dx.doi.org/10.1090/S0273-0979-01-00920-X">Random matrix theory over finite fields</a>, 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
%H A038348 Rebekah Ann Gilbert, <a href="https://web.archive.org/web/20220122155233/https://faculty.math.illinois.edu/~rgilber1/AFineRediscovery_Gilbert.pdf">A Fine Rediscovery</a>, 2014.
%H A038348 Amrik Singh Nimbran and Paul Levrie, <a href="https://repository.uantwerpen.be/docstore/d:irua:21961">Series of the form Sum {a_n*binomial(2n, n)}</a>, Math. Student (2023) Vol. 92, Nos. 3-4, 155-173. See pp. 10, 16.
%F A038348 a(n) = A036469(n) - a(n-1) = Sum_{k=0..n} (-1)^k*A036469(n-k). - _Vladeta Jovovic_, Sep 10 2003
%F A038348 a(n) = A000009(n) + a(n-2). - _Vladeta Jovovic_, Feb 10 2004
%F A038348 G.f.: 1/((1-x^2)*Product_{j>=1} (1 - x^(2*j-1))). - _Emeric Deutsch_, Feb 22 2006
%F A038348 From _Vaclav Kotesovec_, Aug 16 2015: (Start)
%F A038348 a(n) ~ (1/2) * A036469(n).
%F A038348 a(n) ~ 3^(1/4) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)). (End)
%F A038348 Euler transform of the sequence [1, 1, period(1, 0)] (A266591). - _Georg Fischer_, Dec 04 2020
%e A038348 From _Gus Wiseman_, Sep 23 2019: (Start)
%e A038348 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:
%e A038348 (1) (2) (3) (4) (5) (6) (7)
%e A038348 (11) (21) (31) (32) (42) (43)
%e A038348 (111) (211) (41) (51) (52)
%e A038348 (1111) (311) (321) (61)
%e A038348 (2111) (411) (421)
%e A038348 (11111) (3111) (511)
%e A038348 (21111) (3211)
%e A038348 (111111) (4111)
%e A038348 (31111)
%e A038348 (211111)
%e A038348 (1111111)
%e A038348 (End)
%p A038348 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
%t A038348 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 *)
%o A038348 (SageMath) # uses[EulerTransform from A166861]
%o A038348 def g(n): return n % 2 if n > 2 else 1
%o A038348 a = EulerTransform(g)
%o A038348 print([a(n) for n in range(48)]) # _Peter Luschny_, Dec 04 2020
%Y A038348 Cf. A067588, A090867, A116674, A173305.
%Y A038348 Cf. A000009, A007360, A051424, A259936, A266591, A302569, A306200.
%K A038348 nonn
%O A038348 0,3
%A A038348 _N. J. A. Sloane_