|
| |
|
|
A035959
|
|
Number of partitions of n in which no parts are multiples of 5.
|
|
12
| |
|
|
1, 1, 2, 3, 5, 6, 10, 13, 19, 25, 34, 44, 60, 76, 100, 127, 164, 205, 262, 325, 409, 505, 628, 769, 950, 1156, 1414, 1713, 2081, 2505, 3026, 3625, 4352, 5192, 6200, 7364, 8756, 10357, 12258, 14450, 17034, 20006, 23500, 27510, 32200, 37582, 43846, 51022
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| Also number of partitions with at most 4 parts of size 1 and differences between parts at distance 6 are greater than 1.
Also number of partitions of n where no part appears more than four times.
Case k=7,i=5 of Gordon Theorem.
|
|
|
REFERENCES
| G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.
Watson, G. N.; Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128. See the expression Y = B/C in the notation of p. 106. [Added by N. J. A. Sloane, Nov 13 2009]
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
GDZ, Digitized volumes of Crelle [Added by N. J. A. Sloane, Nov 13 2009]
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
|
|
|
FORMULA
| G.f.: Prod_{j=1, oo} (1 + x^j + x^2j + x^3j + x^4j) - Jon Perry (perry(AT)globalnet.co.uk), Mar 30 2004
G.f.: Prod_{n>0, n==1, 2, 3, 4 mod 5} 1/(1-q^n).
Given g.f. A(x) then B(x) = x * A(x^3)^2 satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^3 + v^3 - u*v - 5*u^2*v^2. - Michael Somos, May 28 2006
Given g.f. A(x) then B(x) = x * A(x^3)^2 satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v + 5*v^2*(u + w) - (u^2 + u*w + w^2). - Michael Somos, May 28 2006
Euler transform of period 5 sequence [ 1, 1, 1, 1, 0, ...]. - Michael Somos, May 28 2006
G.f. is product k>0 P5(x^k) where P5 is 5th cyclotomic polynomial.
|
|
|
EXAMPLE
| 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 6*x^5 + 10*x^6 + 13*x^7 + 19*x^8 + ...
|
|
|
MATHEMATICA
| max = 47; f[x_] := (x^5-1)/(x-1); g[x_] := Product[f[x^k], {k, 1, max}]; CoefficientList[ Series[g[x], {x, 0, max}], x] (* From Jean-François Alcover, Nov 29 2011, after Michael Somos *)
|
|
|
PROG
| (PARI) {a(n) = if( n<0, 0, polcoeff( eta(x^5 + x * O(x^n)) / eta(x + x * O(x^n)), n))} /* Michael Somos, May 28 2006 */
(Haskell)
a035959 = p a047201_list where
p _ 0 = 1
p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Dec 17 2011
|
|
|
CROSSREFS
| Cf. A000726, A001935, A000009, A061198, A061199.
Cf. A047201.
Sequence in context: A035953 A195054 A087750 * A036801 A035966 A035974
Adjacent sequences: A035956 A035957 A035958 * A035960 A035961 A035962
|
|
|
KEYWORD
| nonn,easy,nice,changed
|
|
|
AUTHOR
| Olivier Gerard (olivier.gerard(AT)gmail.com)
|
| |
|
|
