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A003105 Schur's 1926 partition theorem: number of partitions of n into parts 6n+1 or 6n-1.
(Formerly M0254)
42
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; text; 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
Coefficients of replicable function number 72e. - N. J. A. Sloane, Jun 10 2015
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.) - N. Sato, Jul 20 2005
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution of A262928 and A261612. - Vaclav Kotesovec, Jan 13 2017
Convolution of A109702 and A109701. - Vaclav Kotesovec, Jan 21 2017
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.
H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
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
Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 201 terms from R. Zumkeller)
K. Alladi and B. Gordon, Generalizations of Schur's partition theorem, Manuscr. Math. 79 (1993) 113-126.
K. Alladi and B. Gordon, Schur's partition theorem, companions, refinements and generalizations, Trans. Amer. Math. Soc. 347 (1995) 1591-1608.
G. E. Andrews, K. Alladi, B. Gordon, Generalizations and refinements of a partition theorem of Göllnitz, J. Reine Angew. Math. 460 (1995) 165-188.
D. M. Bressoud, A combinatorial proof of Schur's 1926 partition theorem, Proc. Amer, Math. Soc. 79 (1980) 338-340.
N. Chair, Partition identities from Partial Supersymmetry, arXiv:hep-th/0409011, 2004.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
H. Göllnitz, Partitionen mit Differenzenbedingungen, J. Reine Angew. Math. Vol. 225 (1967), 154-190.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 12.
Padmavathamma, R. Raghavendra and B. M. Chandrashekara, A new bijective proof of a partition theorem of K. Alladi, Discrete Math., 237 (2004), 125-128.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Schur's Partition Theorem
James J. Y. Zhao, A Bijective Proof of the Alladi-Andrews-Gordon Partition Theorem, Elect. J. Combin, Volume 22, Issue 1 (2015) Paper #P1.68.
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, 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 f(x, x^2) / f(-x^3) = f(-x^6) / f(-x, -x^5) in powers of x where f() is Ramanujan theta function. - Michael Somos, Jul 05 2014
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(q) = (A(q^12) / q)^4 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u*v^4 + (1 - u^3) * v^3 + 6*u^2*v^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
a(n) ~ Pi*sqrt(2) / sqrt(3*(12*n-1)) * BesselI(1, Pi*sqrt(12*n-1) / (3*sqrt(6))) ~ exp(Pi*sqrt(2*n)/3) / (2^(5/4) * sqrt(3) * n^(3/4)) * (1 - (9/(8*Pi) + Pi/36)/sqrt(2*n) + (5 - 135/(4*Pi^2) + Pi^2/81)/(64*n)). - Vaclav Kotesovec, Aug 23 2015, extended Jan 09 2017
a(n) = (1/n)*Sum_{k=1..n} A186099(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 21 2017
EXAMPLE
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 + 2*q^59 + 2*q^71 + 3*q^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
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] (* Jean-François Alcover, Dec 01 2011, after Vladeta Jovovic *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] / QPochhammer[ -x^3, x^3], {x, 0, n}]; (* Michael Somos, Jul 05 2014 *)
nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 3] != 0, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 13 2017 *)
nmax = 63; kmax = nmax/6;
s = Flatten[{Range[0, kmax]*6 + 1}~Join~{Range[kmax]*6 - 1}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Jul 31 2020 *)
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
(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), ", "))
/* Paul D. Hanna, Feb 17 2013 */
CROSSREFS
Cf. A186099 (log).
Sequence in context: A029094 A262950 A227398 * A240856 A081166 A173910
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Jun 08 2003
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)