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A015128
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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.
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19
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1, 2, 4, 8, 14, 24, 40, 64, 100, 154, 232, 344, 504, 728, 1040, 1472, 2062, 2864, 3948, 5400, 7336, 9904, 13288, 17728, 23528, 31066, 40824, 53408, 69568, 90248, 116624, 150144, 192612, 246256, 313808, 398640, 504886, 637592, 802936, 1008448
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
Also the number of jagged partitions of n.
Euler transform of period 2 sequence [2,1,...].
According to Ramanujan (1913) a(n) is close to (cosh(x)-sinh(x)/x)/(4*n) where x=pi*sqrt(n).
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
Number of partitions of n where there are two kinds of odd parts. [Joerg Arndt, Jul 30 2011]
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
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.
a(8n+7) is divisible by 64 (from Fortin/Jacob/Mathieu paper).
Smallest sequence of even numbers (except a(0)) which is the Euler transform of a sequence of positive integers. - Franklin T. Adams-Watters, Oct 16 2006
Convolution of A000041 and A000009. - Vladeta Jovovic (vladeta(AT)eunet.rs), Nov 26 2002
Equals A022567 convolved with A035363 [From Gary W. Adamson, Jun 09 2009]
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, Jul 05 2009]
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REFERENCES
| J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 103.
S. Corteel, Particle seas and basic hypergeometric series, Advances Appl. Math., 31 (2003), 199-214.
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.
S. Corteel and J. Lovejoy, Overpartitions, Trans. Amer. Math. Soc., 356 (2004), 1623-1635.
Drake, Brian, Limits of areas under lattice paths. Discrete Math. 309 (2009), no. 12, 3936-3953.
A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, in preparation.
B. Kim, A short note on the overpartition function, Discr. Math., 309 (2009), 2528-2532.
B. Kim, Overpartition pairs modulo powers of 2, Discrete Math., 311 (2011), 835-840.
J. Lovejoy, Gordon's theorem for overpartitions, J. Combin. Theory A 103 (2003), 393-401.
K. Mahlburg, The overpartition function modulo small powers of 2, Discr. Math., 286 (2004), 263-267.
J. R. Newman, The World of Mathematics, Simon and Schuster, 1956, Vol. I p. 372.
I. Pak, Partition bijections, a survey, Ramanujan J., to appear.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
N. Chair, Partition identities from Partial Supersymmetry
J.-F. Fortin, P. Jacob and P. Mathieu, Jagged partitions
F. Garvan, Table of a(n) for n=1..10000
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| G.f.: prod(m>=1, (1+q^m)/(1-q^m) ).
G.f.: 1 / (sum(m=-inf..inf, (-q)^(m^2) ) = 1/theta_4(q).
G.f.: 1 / prod(m=1..inf, (1-q^(2*m)) * ( 1-q^(2*m-1))^2 ).
G.f.: exp( Sum_{n>=1} 2*x^(2*n-1)/(1 - x^(2*n-1))/(2*n-1) ). [From Paul D. Hanna, Aug 06 2009]
G.f.: exp( sum(n>=1, (sigma(2*n)-sigma(n))*x^n/n ) ). [Joerg Arndt, Jul 30 2011]
G.f.: prod(n>=0, theta_3(q^(2^n))^(2^n) ). [Joerg Arndt, Aug 03 2011]
Expansion of eta(q^2) / eta(q)^2 in powers of q. - Michael Somos, Jan 04 2011
Expansion of 1 / phi(-q) in powers of q where phi() is a Ramanujan theta function. - - Michael Somos, Nov 01 2008
Recurrence: a(n) = 2*Sum[m>=1, (-1)^(m+1) * a(n-m^2)].
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
G.f. : product(i=1, oo, (1+x^i)^A001511(2i)) (see A000041) - Jon Perry, Jun 06 2004
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
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
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
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
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EXAMPLE
| 1 + 2*q + 4*q^2 + 8*q^3 + 14*q^4 + 24*q^5 + 40*q^6 + 64*q^7 + 100*q^8 + ...
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MAPLE
| mul((1+x^n)/(1-x^n), n=1..256);
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PROG
| (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 */
(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)} /* Paul D. Hanna, Aug 06 2009 */
(PARI) N=66; x='x+O('x^N); /* that many terms */
gf=exp(sum(n=1, N, (sigma(2*n)-sigma(n))*x^n/n));
Vec(gf) /* show terms */ /* Joerg Arndt, Jul 30 2011 */
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CROSSREFS
| Convolution inverse of A002448. A004402(n) = (-1)^n * a(n).
A022567, A035363 [From Gary W. Adamson, Jun 09 2009]
Sequence in context: A069252 A069253 A004402 * A123655 A084683 A118544
Adjacent sequences: A015125 A015126 A015127 * A015129 A015130 A015131
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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