Search: id:A015128
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%I A015128
%S A015128 1,2,4,8,14,24,40,64,100,154,232,344,504,728,1040,1472,2062,2864,3948,
%T A015128 5400,7336,9904,13288,17728,23528,31066,40824,53408,69568,90248,116624,
%U A015128 150144,192612,246256,313808,398640,504886,637592,802936,1008448
%N A015128 Number of overpartitions of n: an overpartition of n is an ordered sequence 
               of nonincreasing integers that sum to n, where the first occurrence 
               of each integer may be overlined.
%C A015128 Also the number of jagged partitions of n.
%C A015128 Euler transform of period 2 sequence [2,1,...].
%C A015128 According to Ramanujan (1913) a(n) is close to (cosh(x)-sinh(x)/x)/(4n) 
               where x=pi*sqrt(n).
%C A015128 Number of partitions of 2n with all odd parts occurring with even multiplicities. 
               There is no restriction on the even parts. Cf. A006950, A046682. 
               - Mamuka Jibladze (jib(AT)rmi.acnet.ge), Sep 05 2003
%C A015128 Coincides with the sequence of numbers of nilpotent conjugacy classes 
               in the Lie algebras sp(n), n=0,1,2,3,... (the case n=0 being degenerate). 
               A006950, this sequence and A000041 together cover the nilpotent conjugacy 
               classes in the classical A,B,C,D series of Lie algebras. - Alexander 
               Elashvili, Sep 08 2003
%C A015128 Also, number of 01-partitions of n. A 01-partition of n is a weakly decreasing 
               sequence of m nonnegative integers n(i) such that sum(n(i))=n, n(m)>
               0, n(j)>=n(j+1)-1 and n(j)>=n(j+2). They are special cases of jagged 
               partitions.
%C A015128 a(8n+7) is divisible by 64 (from Fortin/Jacob/Mathieu paper).
%C A015128 Smallest sequence of even numbers (except a(0)) which is the Euler transform 
               of a sequence of positive integers. - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), 
               Oct 16 2006
%C A015128 Equals A022567 convolved with A035363 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Jun 09 2009]
%C A015128 Equals the infinite product [1,2,2,2,...] * [1,0,2,0,2,0,2,...] * [1,
               0,0,2,0,0,2,0,0,2,...] * ... [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Jul 05 2009]
%D A015128 G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. 
               Math., 7 (No. 4, 1998), pp. 343-359.
%D A015128 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", 
               Springer-Verlag, p. 103.
%D A015128 S. Corteel, Particle seas and basic hypergeometric series, Advances Appl. 
               Math., 31 (2003), 199-214.
%D A015128 S. Corteel and J. Lovejoy, Frobenius partitions and the combinatorics 
               of Ranaujan's {}_1 psi_1 summation, J. Combin. Theory A 97 (2002), 
               177-183.
%D A015128 S. Corteel and J. Lovejoy, Overpartitions, Trans. Amer. Math. Soc., 356 
               (2004), 1623-1635.
%D A015128 K. Mahlburg, The overpartition function modulo small powers of 2, Discr. 
               Math., 286 (2004), 263-267.
%D A015128 A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring 
               at most thrice, in preparation.
%D A015128 J. Lovejoy, Gordon's theorem for overpartitions, J. Combin. Theory A 
               103 (2003), 393-401.
%D A015128 J. R. Newman, The World of Mathematics, Simon and Schuster, 1956, Vol. 
               I p. 372.
%D A015128 I. Pak, Partition bijections, a survey, Ramanujan J., to appear.
%H A015128 T. D. Noe, <a href="b015128.txt">Table of n, a(n) for n=0..1000</a>
%H A015128 G. Almkvist, <a href="http://www.expmath.org/restricted/7/7.4/almkvist.ps">
               Asymptotic formulas and generalized Dedekind sums</a>.
%H A015128 N. Chair, <a href="http://arXiv.org/abs/hep-th/0409011">Partition identities 
               from Partial Supersymmetry</a>
%H A015128 J.-F. Fortin, P. Jacob and P. Mathieu, <a href="http://arXiv.org/abs/
               math.CO/0310079">Jagged partitions</a>
%F A015128 G.f.: Product_{m=1..inf} (1+q^m)/(1-q^m); also (Sum (-q)^(m^2), m = -inf 
               .. inf )^(-1).
%F A015128 Taylor series for 1/theta_4.
%F A015128 Product expansion is 1 / product_{m=1..inf} (1-q^(2m)) * ( 1-q^(2m-1))^2.
%F A015128 Convolution of A000041 and A000009. - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Nov 26 2002
%F A015128 Recurrence: a(n) = 2*Sum[m>=1, (-1)^(m+1) * a(n-m^2)].
%F A015128 a(n) = (1/n)*Sum_{k=1..n} (sigma(2*k)-sigma(k))*a(n-k). - Vladeta Jovovic 
               (vladeta(AT)eunet.rs), Dec 05 2004
%F A015128 G.f. : product(i=1, oo, (1+x^i)^A001511(2i)) (see A000041) - Jon Perry 
               (perry(AT)globalnet.co.uk), Jun 06 2004
%F A015128 Expansion of eta(q)^2 / eta(q^2) in powers of q. - - Michael Somos Nov 
               01 2008
%F A015128 Expansion of 1 / phi(-q) in powers of q where phi() is a Ramanujan theta 
               function. - - Michael Somos Nov 01 2008
%F A015128 G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = w^4 
               * (u^4 + v^4) - 2 * u^2 * v^6. - Michael Somos Nov 01 2008
%F A015128 G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, 
               u3, u6) = u6^3 * (u1^2 + u3^2) - 2 * u1 * u2 * u3^3. - Michael Somos 
               Nov 01 2008
%F A015128 G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, 
               u3, u6) = u2^3 * (u3^2 - 3 * u1^2) + 2 * u1^3 * u3 * u6. - Michael 
               Somos Nov 01 2008
%F A015128 G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 32^(-1/
               2) (t / i)^(-1/2) g(t) where q = exp(2 pi i t) and g() is g.f. for 
               A106507. - Michael Somos Nov 01 2008
%F A015128 G.f.: exp( Sum_{n>=1} 2*x^(2n-1)/(1 - x^(2n-1))/(2n-1) ). [From Paul 
               D. Hanna (pauldhanna(AT)juno.com), Aug 06 2009]
%e A015128 1 + 2*q + 4*q^2 + 8*q^3 + 14*q^4 + 24*q^5 + 40*q^6 + 64*q^7 + 100*q^8 
               + ...
%o A015128 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 
               + A) / eta(x + A)^2, n))} /* Michael Somos Nov 01 2008 */
%o A015128 (PARI) {a(n)=polcoeff(exp(sum(m=1,n\2+1,2*x^(2*m-1)/(1-x^(2*m-1)+x*O(x^n))/
               (2*m-1))),n)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Aug 06 
               2009]
%Y A015128 Convolution inverse of A002448. A004402(n) = (-1)^n * a(n).
%Y A015128 A022567, A035363 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 09 
               2009]
%Y A015128 Sequence in context: A069252 A069253 A004402 this_sequence A123655 A084683 
               A118544
%Y A015128 Adjacent sequences: A015125 A015126 A015127 this_sequence A015129 A015130 
               A015131
%K A015128 nonn,easy,nice
%O A015128 0,2
%A A015128 N. J. A. Sloane (njas(AT)research.att.com).

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