|
| |
|
|
A003105
|
|
Number of partitions of n into parts 6n+1 or 6n-1.
(Formerly M0254)
|
|
13
| |
|
|
1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 23, 26, 30, 34, 38, 42, 47, 53, 60, 67, 74, 82, 91, 102, 114, 126, 139, 153, 169, 187, 207, 228, 250, 274, 301, 331, 364, 399, 436, 476, 520, 569, 622, 679, 739, 804, 875, 953, 1038, 1128, 1224, 1327
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,6
|
|
|
COMMENTS
| There are many (at least 8) equivalent definitions of this sequence (besides the comments below, see also Schur, Alladi, Andrews). - N. J. A. Sloane, Jun 17 2011
McKay-Thompson series of class 72e for the Monster group.
Also number of partitions of n into odd parts in which no part appears more than twice, cf. A070048 and A096938. - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 18 2005
Also number of partitions of n into distinct parts congruent to 1 or 2 modulo 3. (Follows from second G.F.) - Naoki Sato (nsato7(AT)yahoo.ca), Jul 20 2005
|
|
|
REFERENCES
| K. Alladi, Refinements of Rogers-Ramanujan type identities. In Special Functions, q-Series and Related Topics (Toronto, ON, 1995), 1-35, Fields Inst. Commun., 14, Amer. Math. Soc., Providence, RI, 1997.
G. E. Andrews, Schur's theorem, partitions with odd parts and the Al-Salam-Carlitz polynomials. In q-Series From a Contemporary Perspective (South Hadley, MA, 1998), 45-56, Contemp. Math., 254, Amer. Math. Soc., Providence, RI, 2000.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
I. Schur, Zur Additiven Zahlentheorie, Ges. Abh., Vol. 2, Springer, pp. 43-50.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
| R. Zumkeller, Table of n, a(n) for n = 0..200
N. Chair, Partition identities from Partial Supersymmetry
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Index entries for McKay-Thompson series for Monster simple group
|
|
|
FORMULA
| G.f.: 1/Product_{k>=0} (1-x^(6*k+1))*(1-x^(6*k+5)) = Product_{k>=0} (1+x^(3*k+1))*(1+x^(3*k+2)) = 1/Product_{k>=0} (1-x^k+x^(2*k)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jun 08 2003
Expansion of q^(1/12)eta(q^2)eta(q^3)/(eta(q)eta(q^6)) in powers of q.
Euler transform of period 6 sequence [1, 0, 0, 0, 1, 0, ...]. - Michael Somos, Jan 09 2005
Given g.f. A(x), then B(x)=(A(x^12)/x)^4 satisfies 0=f(B(x), B(x^2)) where f(u, v)=uv^4+(1-u^3)v^3+6u^2v^2+(u^4-u)v+u^3 - Michael Somos, Jan 09 2005
|
|
|
EXAMPLE
| T72e = 1/q + q^11 + q^23 + q^35 + q^47 + 2q^59 + 2q^71 + 3q^83 + ...
|
|
|
MAPLE
| with(combinat);
A:=proc(n) local i, j, t3, t2, t1;
t2:=0;
t1:=firstpart(n);
for j from 1 to numbpart(n)+2 do
t3:=1;
for i from 1 to nops(t1) do
if (t1[i] mod 6) <> 1 and (t1[i] mod 6) <> 5 then t3:=0; fi;
od;
if t3=1 then t2:=t2+1; fi;
if nops(t1) = 1 then RETURN(t2); fi;
t1:=nextpart(t1);
od;
end;
# - brute-force Maple program from N. J. A. Sloane, Jun 17 2011
|
|
|
MATHEMATICA
| max = 63; f[x_] := 1/Product[1 - x^k + x^(2k), {k, 0, max}]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* From Jean-François Alcover, Dec 01 2011, after Vladeta Jovovic *)
|
|
|
PROG
| (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)*eta(x^3+A)/eta(x+A)/eta(x^6+A), n))} /* Michael Somos Jan 09 2005 */
(Haskell)
a003105 n = p 1 n where
p k m | m == 0 = 1 | m < k = 0 | otherwise = q k (m-k) + p (k+2) m
q k m | m == 0 = 1 | m < k = 0 | otherwise = p (k+2) (m-k) + p (k+2) m
-- Reinhard Zumkeller, Nov 12 2011
|
|
|
CROSSREFS
| Cf. A000041, A001651, A003114, A000726, A109697, A132462, A132463.
Sequence in context: A125059 A029112 A029094 * A081166 A173910 A036846
Adjacent sequences: A003102 A003103 A003104 * A003106 A003107 A003108
|
|
|
KEYWORD
| nonn,nice
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Herman P. Robinson
|
|
|
EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Jun 08 2003
|
| |
|
|
