|
Search: id:A026832
|
|
|
| A026832 |
|
Number of partitions of n into distinct parts, the least being odd. |
|
+0
4
|
|
| 1, 0, 2, 1, 2, 2, 4, 4, 5, 6, 8, 10, 12, 14, 18, 21, 24, 30, 36, 42, 50, 58, 68, 80, 93, 108, 126, 146, 168, 194, 224, 256, 294, 336, 384, 439, 500, 568, 646, 732, 828, 938, 1060, 1194, 1348, 1516, 1704, 1916, 2149, 2408, 2698, 3018, 3372
(list; graph; listen)
|
|
|
OFFSET
|
1,3
|
|
|
COMMENT
|
Fine's numbers L(n).
Also number of partitions of n such that if k is the largest part, then k occurs an odd number of times and each of the numbers 1,2,...,k-1 occurs at least once. Example: a(7)=4 because we have [3,2,1,1],[2,2,2,1],[2,1,1,1,1,1] and [1,1,1,1,1,1,1] - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 29 2006
|
|
REFERENCES
|
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 56, Eq. (26.28).
|
|
FORMULA
|
G.f.: Sum_{k>=1} ((-1)^(k+1)*(-1+Product_{i>=k} (1+x^i))). - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 26 2003
G.f.: Sum_{ k >= 1 } x^(k*(k+1)/2)/((1+x^k)*Product_{i=1..k} (1-x^i) ). - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 10 2004
(1 + Sum_{n >= 1} a(n)q^n )*(1 + 2 Sum_{m>=1} (-1)^m*q^(m^2)) = Sum_{n >= 1} (-1)^n*q^((3*n^2+n)/2)/(1+q^n). [Fine]
G.f.=sum(x^(2k-1)*product(1+x^j, j=2k..infinity), k=1..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 29 2006
|
|
EXAMPLE
|
a(7)=4 because we have [7],[6,1],[4,3] and [4,2,1].
|
|
MAPLE
|
g:=sum(x^(2*k-1)*product(1+x^j, j=2*k..60), k=1..60): gser:=series(g, x=0, 55); seq(coeff(gser, x^n), n=1..53); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 29 2006
|
|
CROSSREFS
|
Cf. A026804, A026805, A026807, A092265, A096661, A097042.
Sequence in context: A133770 A163373 A117193 this_sequence A089408 A079318 A050315
Adjacent sequences: A026829 A026830 A026831 this_sequence A026833 A026834 A026835
|
|
KEYWORD
|
nonn,nice
|
|
AUTHOR
|
Clark Kimberling (ck6(AT)evansville.edu)
|
|
EXTENSIONS
|
More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 29 2006
|
|
|
Search completed in 0.002 seconds
|
