|
| |
|
|
A002513
|
|
Expansion of product 1/((1-x^(2k))^2 (1-x^(2k-1))), k>0.
(Formerly M2354 N0931)
|
|
1
| |
|
|
1, 1, 3, 4, 9, 12, 23, 31, 54, 73, 118, 159, 246, 329, 489, 651, 940, 1242, 1751, 2298, 3177, 4142, 5630, 7293, 9776, 12584, 16659, 21320, 27922, 35532, 46092, 58342, 75039, 94503, 120615, 151173, 191611, 239060, 301086, 374026, 468342, 579408
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| Expansion of q^(1/8)/(eta(q)eta(q^2)) in powers of q.
Euler transform of period 2 sequence [1,2,...].
For a real polynomial equation of degree n, a(n) is the number of possibilities for the roots to be real and unequal, real and equal (in various combinations), or simple or multiple complex conjugates. For example, a(3)=4 because we can have: three equal roots, two equal roots, three distinct real roots and two complex roots (see the Monthly Problem reference). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 22 2005
Number of partitions of n, the even parts being of two kinds. E.g. a(4)=9 because we have 4, 4', 3+1, 2+2, 2+2', 2'+2', 2+1+1, 2'+1+1, 1+1+1+1. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 22 2005
|
|
|
REFERENCES
| Newman, Morris; Construction and application of a class of modular functions. II. Proc. London Math. Soc. (3) 9 1959 373-387.
Problem E2055, Amer. Math. Monthly, 75 (1968), 188; 76 (1969), 194.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
|
|
|
FORMULA
| Given g.f. A(x), then B(x)=A(x)^8/x satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=16v^4+v^3w+256uv^3+16uv^2w-u^2w^2. - Michael Somos Apr 3 2005
G.f.: Product_{k>0} 1/((1-x^(2k))^2 (1-x^(2k-1))).
|
|
|
MATHEMATICA
| max = 50; f[x_] := Product[ 1/((1-x^(2 k))^2*(1-x^(2k-1))), {k, 1, Ceiling[max/2]} ]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* From Jean-François Alcover, Nov 04 2011 *)
|
|
|
PROG
| (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( 1/eta(x+A)/eta(x^2+A), n))} /* Michael Somos Nov 10 2005 */
|
|
|
CROSSREFS
| Sequence in context: A025613 A097063 A026476 * A034418 A034421 A029448
Adjacent sequences: A002510 A002511 A002512 * A002514 A002515 A002516
|
|
|
KEYWORD
| nonn,easy,nice
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
|
|
|
EXTENSIONS
| More terms and information from Michael Somos, Mar 23 2003
|
| |
|
|
