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A006950 G.f.: Product_{k>0} (1 + x^(2*k - 1)) / (1 - x^(2*k)).
(Formerly M0524)
49
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; text; 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, 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

Poincaré series (or Molien series) for symmetric invariants in F_2(b_1, b_2, ... b_n) ⊗ 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 restriction 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, Feb 03 2005

Equals polcoeff inverse of A010054 with alternate signs. - Gary W. Adamson, 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 column 1 of A195836, also column 2 of the square array A195825. - Omar E. Pol, Oct 09 2011

Since this is also column 2 of A195825 so the sequence contains only one plateau [1, 1, 1] of level 1 and length 3. For more information see A210843. - Omar E. Pol, Jun 27 2012

Convolution of A035363 and A000700. - Vaclav Kotesovec, Aug 17 2015

REFERENCES

A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 108.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

N. Chair, Partition identities from Partial Supersymmetry, arXiv:hep-th/0409011, 2004.

Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.

Luca Ferrari, Schröder partitions, Schröder tableaux and weak poset patterns, arXiv:1606.06624 [math.CO], 2016. Mentions this sequence.

Victor S. Miller, Counting Matrices that are Squares, arXiv:1606.09299 [math.GR], 2016.

M. Somos, Introduction to Ramanujan theta functions

Maxie D. Schmidt, Exact Formulas for the Generalized Sum-of-Divisors Functions, arXiv:1705.03488 [math.NT], 2017. See Example 4.2 p. 13.

Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

M. P. Zaletel and R. S. K. Mong, Exact Matrix Product States for Quantum Hall Wave Functions, arXiv preprint arXiv:1208.4862 [cond-mat.str-el], 2012. - From N. J. A. Sloane, Dec 25 2012

FORMULA

a(n) = (1/n)*Sum_{k=1..n} (-1)^(k+1)*A002129(k)*a(n-k), n>1, a(0)=1. - Vladeta Jovovic, Feb 05 2002

G.f.: 1/Sum_{k>=0} (-x)^(k*(k+1)/2). - Vladeta Jovovic, Sep 22 2002 [corrected by Vaclav Kotesovec, Aug 17 2015]

a(n) = A059777(n-1)+A059777(n), n>0. - Vladeta Jovovic, Sep 22 2002

G.f.: Product (1+x^m)^(if A001511(m)>1, A001511(m)-1 else A001511(m)); m=1..inf. - Jon Perry, 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 ). - Paul D. Hanna, Jul 22 2009

a(n) ~ (8*n+1) * cosh(sqrt(8*n-1)*Pi/4) / (16*sqrt(2)*n^2) - sinh(sqrt(8*n-1)*Pi/4) / (2*Pi*n^(3/2)) ~ exp(Pi*sqrt(n/2))/(4*sqrt(2)*n) * (1 - (2/Pi + Pi/16)/sqrt(2*n) + (3/16 + Pi^2/1024)/n). - Vaclav Kotesovec, Aug 17 2015, extended Jan 09 2017

EXAMPLE

G.f. = 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 + ...

G.f. = q^-1 + q^7 + q^15 + 2*q^23 + 3*q^31 + 4*q^39 + 5*q^47 + 7*q^55 + 10*q^63 + ...

MAPLE

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

      b(n, i-1)+`if`(i>n, 0, b(n-i, i-irem(i, 2)))))

    end:

a:= n-> b(n, n):

seq(a(n), n=0..50);  # Alois P. Heinz, Jan 06 2013

MATHEMATICA

CoefficientList[ Series[ Product[(1 + x^(2k - 1))/(1 - x^(2k)), {k, 25}], {x, 0, 50}], x] (* Robert G. Wilson v, Jun 28 2012 *)

CoefficientList[Series[x*QPochhammer[-1/x, x^2] / ((1+x)*QPochhammer[x^2, x^2]), {x, 0, 50}], x] (* Vaclav Kotesovec, Aug 17 2015 *)

CoefficientList[Series[2*(-x)^(1/8) / EllipticTheta[2, 0, Sqrt[-x]], {x, 0, 50}], x] (* Vaclav Kotesovec, Aug 17 2015 *)

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)} \\ Paul D. Hanna, Jul 22 2009

(GWbasic)' A program with two A-numbers (Note that here A000217 are the generalized hexagonal numbers):

10 Dim A000217(100), A057077(100), a(100): a(0)=1

20 For n = 1 to 51: For j = 1 to n

30 If A000217(j) <= n then a(n) = a(n) + A057077(j-1)*a(n - A000217(j))

40 Next j: Print a(n-1); : Next n

# Omar E. Pol, Jun 10 2012

CROSSREFS

Cf. A015128, A046682, A106459.

Cf. A163203.

Cf. A010054.

Sequence in context: A036034 A280949 A106507 * A052335 A193771 A160333

Adjacent sequences:  A006947 A006948 A006949 * A006951 A006952 A006953

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Warren D. Smith

EXTENSIONS

G.f. and more terms from Vladeta Jovovic, Feb 05 2002

STATUS

approved

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Last modified July 26 10:24 EDT 2017. Contains 289835 sequences.