|
| |
|
|
A073252
|
|
Expansion of Product (1+q^(2*m-1))^2, m=1..infinity.
|
|
2
| |
|
|
1, 2, 1, 2, 4, 4, 5, 6, 9, 12, 13, 16, 21, 26, 29, 36, 46, 54, 62, 74, 90, 106, 122, 142, 171, 200, 227, 264, 311, 358, 408, 470, 545, 626, 709, 810, 933, 1062, 1198, 1362, 1555, 1760, 1980, 2238, 2536, 2858, 3205, 3602, 4063, 4560, 5092, 5704, 6400, 7150, 7966
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| Number of partitions of n into distinct odd parts of two kinds. [Joerg Arndt, Jul 30 2011]
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Combinatorial interpretation of sequence: [ X1,X2 ] = 2 strictly increasing sequences (possibly null) of odd positive integers; a(n)=#pairs with sum of entries = n.
McKay-Thompson series of class 48g for the Monster group.
|
|
|
REFERENCES
| T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, q_2^2.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
|
|
|
LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for McKay-Thompson series for Monster simple group
D. Foata and G.-N. Han, Jacobi and Watson Identities Combinatorially Revisited
|
|
|
FORMULA
| G.f.: 1 / (Prod_{k>0} 1 + (-x)^k)^2 = (Prod_{k>0} 1 + x^(2k-1))^2.
Expansion of q^(1/12) * (eta(q^2)^2 / (eta(q) * eta(q^4)))^2 in powers of q.
Expansion of chi(q)^2 = phi(q) / f(-q^2) = f(q) / psi(-q) = (phi(q) / f(q))^2 = (psi(q) / f(-q^4))^2 = (f(-q^2) / psi(-q))^2 = (phi(-q^2) / f(-q))^2 = (f(q) / f(-q^2))^2 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 4 sequence [2, -2, 2, 0, ...].
G.f.: ( prod_{k>0}, 1 + x^(2*k - 1) )^2.
Convolution square of A000700. A022597(n) = (-1)^n * a(n).
|
|
|
EXAMPLE
| a(4)=4:[ (1),(3) ],[ (3),(1) ],[ (),(1,3) ],[ (1,3),() ]
1 + 2*x + x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 5*x^6 + 6*x^7 + 9*x^8 + 12*x^9 + ...
T48g = 1/q + 2*q^11 + q^23 + 2*q^35 + 4*q^47 + 4*q^59 + 5*q^71 + 6*q^83 +...
|
|
|
PROG
| (PARI) {a(n) = if( n<0, 0, polcoeff( prod( i=1, (1+n)\2, 1 + x^(2*i - 1), 1 + x * O(x^n))^2, n))}
(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / prod( i=1, n, 1 + (-x)^i, 1 + x * O(x^n))^2, n))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x*O(x^n); polcoeff( (eta(x^2 + A)^2 / eta(x + A) / eta(x^4 + A))^2, n))}
|
|
|
CROSSREFS
| Cf. A000700, A022597.
Sequence in context: A023673 A132965 A022597 * A134005 A132320 A076369
Adjacent sequences: A073249 A073250 A073251 * A073253 A073254 A073255
|
|
|
KEYWORD
| nonn,easy
|
|
|
AUTHOR
| Michael Somos, Jul 22, 2002
|
|
|
EXTENSIONS
| Comments from Len Smiley (smiley(AT)math.uaa.alaska.edu).
|
| |
|
|
