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A003105
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Number of partitions of n into parts 6n+1 or 6n-1.
(Formerly M0254)
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16
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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
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OFFSET
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0,6
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COMMENTS
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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, 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
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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REFERENCES
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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).
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LINKS
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R. Zumkeller, Table of n, a(n) for n = 0..200
N. Chair, Partition identities from Partial Supersymmetry
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Schurs Partition Theorem
Index entries for McKay-Thompson series for Monster simple group
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FORMULA
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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, Jun 08 2003
Expansion of chi(-x^3) / chi(-x) in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Mar 04 2012
Expansion of q^(1/12) * eta(q^2) * eta(q^3) / (eta(q) * eta(q^6)) in powers of q. - Michael Somos, Jan 09 2005
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
The logarithmic derivative equals A186099. - Paul D. Hanna, Feb 17 2013
G.f.: exp( Sum_{n>=1} A186099(n) * x^n/n ) where A186099(n) = sum of divisors of n congruent to 1 or 5 mod 6. - Paul D. Hanna, Feb 17 2013
G.f.: exp( Sum_{n>=1} S(n,x) * x^n/n ) where S(n,x) = Sum_{d|n} d*(1-x^d)^(n/d). - Paul D. Hanna, Feb 17 2013
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EXAMPLE
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G.f: A(x) = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + 3*x^9 + 4*x^10 + ...
T72e = 1/q + q^11 + q^23 + q^35 + q^47 + 2q^59 + 2q^71 + 3q^83 + ...
The logarithm of the g.f. begins:
log(A(x)) = x + x^2/2 + x^3/3 + x^4/4 + 6*x^5/5 + x^6/6 + 8*x^7/7 + x^8/8 + x^9/9 + 6*x^10/10 + 12*x^11/11 + x^12/12 +...+ A186099(n)*x^n/n +... [Paul D. Hanna, Feb 17 2013]
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
(PARI)
{S(n, x)=sumdiv(n, d, d*(1-x^d)^(n/d))}
{a(n)=polcoeff(exp(sum(k=1, n, S(k, x)*x^k/k)+x*O(x^n)), n)}
for(n=0, 60, print1(a(n), ", "))
/* From Paul D. Hanna, Feb 17 2013 */
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CROSSREFS
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Cf. A000041, A001651, A003114, A000726, A109697, A132462, A132463.
Cf. A186099 (log).
Sequence in context: A125059 A029112 A029094 * A081166 A173910 A036846
Adjacent sequences: A003102 A003103 A003104 * A003106 A003107 A003108
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane, Herman P. Robinson
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EXTENSIONS
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More terms from Vladeta Jovovic, Jun 08 2003
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STATUS
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approved
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