|
| |
|
|
A006950
|
|
G.f.: Product_{k>0} (1 + x^(2*k - 1)) / (1 - x^(2*k)).
(Formerly M0524)
|
|
30
| |
|
|
1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 21, 28, 35, 43, 55, 70, 86, 105, 130, 161, 196, 236, 287, 350, 420, 501, 602, 722, 858, 1016, 1206, 1431, 1687, 1981, 2331, 2741, 3206, 3740, 4368, 5096, 5922, 6868, 7967, 9233, 10670, 12306, 14193, 16357, 18803, 21581
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,4
|
|
|
COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number of partitions of n in which each even part occurs with even multiplicity. There is no restriction on the odd parts.
Also the number of partitions of n into parts not congruent to 2 mod 4 - James A. Sellers (sellersj(AT)math.psu.edu), Feb 08, 2002
Coincides with the sequence of numbers of nilpotent conjugacy classes in the Lie algebras o(n) of skew-symmetric n X n matrices, n=0,1,2,3,... (the cases n=0,1 being degenerate). This sequence, A015128 and A000041 together cover the nilpotent conjugacy classes in the classical A,B,C,D series of Lie algebras. - Alexander Elashvili, Sep 08 2003
Poincare series (or Molien series) for symmetric invariants in F_2(b_1, b_2, ... b_n) \otimes E(e_1, e_2, ... e_n) with b_i 2-dimensional, e_i one-dimensional and the permutation action of S_n, in the case n=2.
Also the number of partitions of n in which all odd parts occur with multiplicity 1. There is no restricton on the even parts. E.g a(9)=13 because "9=8+1=7+2=6+3=6+2+1=5+4=5+3+1=5+2+2=4+4+1=4+3+2=4+2+2+1= 3+2+2+2=2+2+2+2+1" - Noureddine Chair (n.chair(AT)rocketmail.com), Feb 03 2005
Equals polcoeff inverse of A010054 with alternate signs. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 15 2010]
It appears that this sequence is related to the generalized hexagonal numbers (A000217) in the same way as the partition numbers A000041 are related to the generalized pentagonal numbers A001318. (See the table in comments section of A195825). Conjecture: this is 1 together with the row sums of triangle A195836, also the column 1 of A195836, also the column 2 of the square array A195825. - Omar E. Pol, Oct 09 2011
|
|
|
REFERENCES
| A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 108.
Drake, Brian, Limits of areas under lattice paths. Discrete Math. 309 (2009), no. 12, 3936-3953.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
N. Chair, Partition identities from Partial Supersymmetry
|
|
|
FORMULA
| a(n)=(1/n)*Sum_{k=1..n} (-1)^(k+1)*A002129(k)*a(n-k), n>1, a(0)=1. - Vladeta Jovovic (vladeta(AT)eunet.rs), Feb 05 2002
G.f.: 1/Sum_{k>0} (-x)^(k*(k+1)/2). a(n) = A059777(n-1)+A059777(n), n>0. - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 22 2002
G.f.: Product (1+x^m)^(if A001511(m)>1, A001511(m)-1 else A001511(m)); m=1..inf - Jon Perry (perry(AT)globalnet.co.uk), Apr 15 2005
Expansion of 1 / psi(-x) in powers of x where psi() is a Ramanujan theta function.
Expansion of q^(1/8) * eta(q^2) / (eta(q) * eta(q^4)) in powers of q.
Convolution inverse of A106459. - Michael Somos, Nov 02 2005
G.f.: exp( Sum_{n>=1} [Sum_{d|n} (-1)^(n-d)*d] * x^n/n ). [From Paul D. Hanna (pauldhanna(AT)juno.com), Jul 22 2009]
|
|
|
EXAMPLE
| 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 7*x^7 + 10*x^8 + 13*x^9 + ...
q^-1 + q^7 + q^15 + 2*q^23 + 3*q^31 + 4*q^39 + 5*q^47 + 7*q^55 + 10*q^63 + ...
|
|
|
PROG
| (PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, sumdiv(m, d, (-1)^(m-d)*d)*x^m/m)+x*O(x^n)), n)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Jul 22 2009]
|
|
|
CROSSREFS
| Cf. A015128, A046682, A106459.
Cf. A163203. [From Paul D. Hanna (pauldhanna(AT)juno.com), Jul 22 2009]
Cf. A010054 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 15 2010]
Sequence in context: A014670 A036034 * A106507 A052335 A160333 A174578
Adjacent sequences: A006947 A006948 A006949 * A006951 A006952 A006953
|
|
|
KEYWORD
| nonn,changed
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Warren D. Smith
|
|
|
EXTENSIONS
| G.f. and more terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Feb 05 2002
|
| |
|
|
