|
| |
|
|
A027187
|
|
Number of partitions of n into an odd number of parts, the least being 1; also a(n+1) = number of partitions of n into an even number of parts; also partitions of n+1 in which the greatest part is even.
|
|
15
| |
|
|
1, 0, 1, 1, 3, 3, 6, 7, 12, 14, 22, 27, 40, 49, 69, 86, 118, 146, 195, 242, 317, 392, 505, 623, 793, 973, 1224, 1498, 1867, 2274, 2811, 3411, 4186, 5059, 6168, 7427, 9005, 10801, 13026, 15572, 18692, 22267, 26613, 31602, 37619, 44533, 52815, 62338
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,5
|
|
|
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).
|
|
|
REFERENCES
| N. J. Fine, Problem 4314, Amer. Math. Monthly, Vol. 57, 1950, 421-423.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 8, (7.323) and p. 39, Example 7.
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n = 1..1000
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
|
|
|
FORMULA
| a(n+1)=(A000041(n)+(-1)^n*A000700(n))/2.
a(n+1)=p(n)-p(n-1)+p(n-4)-p(n-9)+... where p(n) is the number of unrestricted partitions of n, A000041. [Fine] - David Callan (callan(AT)stat.wisc.edu), Mar 14 2004
G.f.: A(q) = Sum_{n >= 0} a(n) q^n = 1 + q^2 + q^3 + 3 q^4 + 3 q^5 + 6 q^6 + ...
= Sum_{n >= 0} q^(2n)/(q; q)_{2n}
= ((Prod_{k >= 1} 1/(1-q^k) + (Prod_{k >= 1} 1/(1+q^k))/2 (R. William Gosper (rwg(AT)osots.com), Jun 25 2005)
Also, let B(q) = Sum_{n >= 0} A027193(n) q^n = q + q^2 + 2 q^3 + 2 q^4 + 4 q^5 + 5 q^6 + ...
Then B(q) = Sum_{n >= 0} q^(2n+1)/(q; q)_{2n+1} = ((Prod_{k >= 1} 1/(1-q^k) - (Prod_{k >= 1} 1/(1+q^k))/2.
Also we have the following identity involving 2 X 2 matrices:
Prod_{k >= 1} [ 1/(1-q^2k) q^k/(1-q^2k / q^k/(1-q^2k) 1/(1-q^2k) ]
= [ A(q) B(q) / B(q) A(q) ] (R. William Gosper (rwg(AT)osots.com), Jun 25 2005)
a(2*n)=A046682(2*n), a(2*n+1)=A000701(2*n+1); a(n)=A000041(n)-A027193(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 22 2006
Expansion of (1+phi(q))/(2f(-q)) where phi(),f() are Ramanujan theta functions.
G.f.: (Sum_{k>=0} (-1)^n x^n^2)/(Product_{k>0}(1-x^k)). - Michael Somos Aug 19 2006
|
|
|
EXAMPLE
| a(1) = 1 from the partition 1 = 1; a(2) = 0; a(3) = 1 from 3 = 1+1+1.
|
|
|
PROG
| (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^oo); polcoeff( sum(k=0, sqrtint(n), (-x)^k^2, A)/eta(x+A), n))} /* Michael Somos Aug 19 2006 */
|
|
|
CROSSREFS
| Cf. A027193, A000701, A046682.
Cf. A026838.
Sequence in context: A083751 A034401 A088571 * A056508 A050065 A078477
Adjacent sequences: A027184 A027185 A027186 * A027188 A027189 A027190
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
|
|
|
EXTENSIONS
| This is an example of a sequence where there are two good choices for the offset. Offset 1 is consistent with the first part of the definition. - N. J. A. Sloane (njas(AT)research.att.com), Aug 27 2006
|
| |
|
|
