|
Search: id:A036016
|
|
|
| A036016 |
|
Number of partitions of n into parts not of form 4k+2, 8k, 8k+3 or 8k-3. |
|
+0
1
|
|
| 1, 1, 1, 1, 2, 2, 2, 3, 4, 5, 5, 6, 8, 9, 10, 12, 15, 17, 19, 22, 26, 30, 33, 38, 45, 51, 56, 64, 74, 83, 92, 104, 119, 133, 147, 165, 187, 208, 229, 256, 288, 319, 351, 390, 435, 481, 528, 584, 649, 715, 783, 863, 954, 1047, 1145, 1258, 1385, 1517, 1655, 1812, 1989
(list; graph; listen)
|
|
|
OFFSET
|
0,5
|
|
|
COMMENT
|
Case k=2,i=2 of Gordon/Goellnitz/Andrews Theorem.
Also number of partitions in which no odd part is repeated, with at most one part of size less than or equal to 2 and where differences between adjacent parts are greater than 1 when the larger part is odd and greater than 2 when the larger part is even.
Euler transform of period 8 sequence [1,0,0,1,0,0,1,0,...]. - Michael Somos Jun 28 2004
|
|
REFERENCES
|
G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 114.
G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573. (See Th. 8.)
S.-D. Chen and S.-S. Huang, On the series expansion of the Goellnitz-Gordon continued fraction, Internat. J. Number Theory, 1 (2005), 53-63.
|
|
LINKS
|
Eric Weisstein's World of Mathematics, Goellnitz-Gordon Identities
|
|
FORMULA
|
Let qf(a, q) = Product(1-a*q^j, j=0..infinity); g.f. is 1/(qf(q, q^8)*qf(q^4, q^8)*qf(q^7, q^8)).
|
|
MAPLE
|
M:=100; qf:=(a, q)->mul(1-a*q^j, j=0..M); tS:=1/(qf(q, q^8)*qf(q^4, q^8)*qf(q^7, q^8)); series(%, q, M); seriestolist(%);
|
|
PROGRAM
|
(PARI) a(n)=if(n<0, 0, polcoeff(1/prod(k=1, n, 1-([1, 0, 0, 1, 0, 0, 1, 0][(k-1)%8+1])*x^k, 1+x*O(x^n)), n)) /* Michael Somos Jun 28 2004 */
|
|
CROSSREFS
|
Sequence in context: A076872 A008906 A029074 this_sequence A051918 A163801 A104661
Adjacent sequences: A036013 A036014 A036015 this_sequence A036017 A036018 A036019
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
Olivier Gerard (olivier.gerard(AT)gmail.com)
|
|
|
Search completed in 0.002 seconds
|
