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%I M0254 #136 Oct 27 2023 20:05:29
%S 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,
%T 47,53,60,67,74,82,91,102,114,126,139,153,169,187,207,228,250,274,301,
%U 331,364,399,436,476,520,569,622,679,739,804,875,953,1038,1128,1224,1327
%N Schur's 1926 partition theorem: number of partitions of n into parts 6n+1 or 6n-1.
%C 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
%C Coefficients of replicable function number 72e. - _N. J. A. Sloane_, Jun 10 2015
%C 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
%C 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
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Convolution of A262928 and A261612. - _Vaclav Kotesovec_, Jan 13 2017
%C Convolution of A109702 and A109701. - _Vaclav Kotesovec_, Jan 21 2017
%D 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.
%D 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 H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
%D I. Schur, Zur Additiven Zahlentheorie, Ges. Abh., Vol. 2, Springer, pp. 43-50.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A003105/b003105.txt">Table of n, a(n) for n = 0..10000</a> (first 201 terms from R. Zumkeller)
%H K. Alladi and B. Gordon, <a href="https://doi.org/10.1007/BF02568332">Generalizations of Schur's partition theorem</a>, Manuscr. Math. 79 (1993) 113-126.
%H K. Alladi and B. Gordon, <a href="https://doi.org/10.1090/S0002-9947-1995-1297520-X ">Schur's partition theorem, companions, refinements and generalizations</a>, Trans. Amer. Math. Soc. 347 (1995) 1591-1608.
%H G. E. Andrews, K. Alladi, B. Gordon, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002212471">Generalizations and refinements of a partition theorem of Göllnitz</a>, J. Reine Angew. Math. 460 (1995) 165-188.
%H D. M. Bressoud, <a href="https://doi.org/10.1090/S0002-9939-1980-0565367-X">A combinatorial proof of Schur's 1926 partition theorem</a>, Proc. Amer, Math. Soc. 79 (1980) 338-340.
%H N. Chair, <a href="http://arXiv.org/abs/hep-th/0409011">Partition identities from Partial Supersymmetry</a>, arXiv:hep-th/0409011, 2004.
%H D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%H H. Göllnitz, <a href="https://dx.doi.org/10.1515/crll.1967.225.154">Partitionen mit Differenzenbedingungen</a>, J. Reine Angew. Math. Vol. 225 (1967), 154-190.
%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 12.
%H Padmavathamma, R. Raghavendra and B. M. Chandrashekara, <a href="https://dx.doi.org/10.1016/j.disc.2004.07.006">A new bijective proof of a partition theorem of K. Alladi</a>, Discrete Math., 237 (2004), 125-128.
%H Herman P. Robinson, <a href="/A003116/a003116.pdf">Letter to N. J. A. Sloane, Nov 13 1973</a>.
%H Herman P. Robinson, <a href="/A003116/a003116_1.pdf">Letter to N. J. A. Sloane, Nov 19 1973</a>.
%H Herman P. Robinson, <a href="/A003105/a003105.pdf">Letter to N. J. A. Sloane, Jan 1974</a>.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SchursPartitionTheorem.html">Schur's Partition Theorem</a>
%H James J. Y. Zhao, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i1p68/0">A Bijective Proof of the Alladi-Andrews-Gordon Partition Theorem</a>, Elect. J. Combin, Volume 22, Issue 1 (2015) Paper #P1.68.
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F 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
%F Expansion of chi(-x^3) / chi(-x) in powers of x where chi() is a Ramanujan theta function. - _Michael Somos_, Mar 04 2012
%F 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
%F 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
%F Euler transform of period 6 sequence [1, 0, 0, 0, 1, 0, ...]. - _Michael Somos_, Jan 09 2005
%F 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
%F The logarithmic derivative equals A186099. - _Paul D. Hanna_, Feb 17 2013
%F 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
%F 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
%F 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
%F a(n) = (1/n)*Sum_{k=1..n} A186099(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 21 2017
%e 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 + ...
%e T72e = 1/q + q^11 + q^23 + q^35 + q^47 + 2*q^59 + 2*q^71 + 3*q^83 + ...
%e The logarithm of the g.f. begins:
%e 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
%p with(combinat);
%p A:=proc(n) local i, j, t3, t2, t1;
%p t2:=0;
%p t1:=firstpart(n);
%p for j from 1 to numbpart(n)+2 do
%p t3:=1;
%p for i from 1 to nops(t1) do
%p if (t1[i] mod 6) <> 1 and (t1[i] mod 6) <> 5 then t3:=0; fi;
%p od;
%p if t3=1 then t2:=t2+1; fi;
%p if nops(t1) = 1 then RETURN(t2); fi;
%p t1:=nextpart(t1);
%p od;
%p end;
%p # brute-force Maple program from _N. J. A. Sloane_, Jun 17 2011
%t 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_ *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] / QPochhammer[ -x^3, x^3], {x, 0, n}]; (* _Michael Somos_, Jul 05 2014 *)
%t 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 *)
%t nmax = 63; kmax = nmax/6;
%t s = Flatten[{Range[0, kmax]*6 + 1}~Join~{Range[kmax]*6 - 1}];
%t Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* _Robert Price_, Jul 31 2020 *)
%o (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 */
%o (Haskell)
%o a003105 n = p 1 n where
%o p k m | m == 0 = 1 | m < k = 0 | otherwise = q k (m-k) + p (k+2) m
%o q k m | m == 0 = 1 | m < k = 0 | otherwise = p (k+2) (m-k) + p (k+2) m
%o -- _Reinhard Zumkeller_, Nov 12 2011
%o (PARI)
%o {S(n,x)=sumdiv(n,d,d*(1-x^d)^(n/d))}
%o {a(n)=polcoeff(exp(sum(k=1,n,S(k,x)*x^k/k)+x*O(x^n)),n)}
%o for(n=0,60,print1(a(n),", "))
%o /* _Paul D. Hanna_, Feb 17 2013 */
%Y Cf. A000041, A001651, A003114, A000726, A109389, A109697, A132462, A132463, A285219, A304047.
%Y Cf. A186099 (log).
%K nonn,nice
%O 0,6
%A _N. J. A. Sloane_, _Herman P. Robinson_
%E More terms from _Vladeta Jovovic_, Jun 08 2003
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