%I M0217 N0078 #287 Jan 10 2024 23:55:45

%S 1,1,0,1,1,1,1,1,2,2,2,2,3,3,3,4,5,5,5,6,7,8,8,9,11,12,12,14,16,17,18,

%T 20,23,25,26,29,33,35,37,41,46,49,52,57,63,68,72,78,87,93,98,107,117,

%U 125,133,144,157,168,178,192,209,223,236,255,276,294,312,335,361,385

%N Expansion of Product_{k>=0} (1 + x^(2k+1)); number of partitions of n into distinct odd parts; number of self-conjugate partitions; number of symmetric Ferrers graphs with n nodes.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C Coefficients of replicable function number 96a. - _N. J. A. Sloane_, Jun 10 2015

%C For n >= 1, a(n) is the minimal row sum in the character table of the symmetric group S_n. The minimal row sum in the table corresponds to the one-dimensional alternating representation of S_n. The maximal row sum is in sequence A085547. - Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 15 2003

%C Also the number of partitions of n into parts != 2 and differing by >= 6 with strict inequality if a part is even. [Alladi]

%C Let S be the set formed by the partial sums of 1+[2,3]+[2,5]+[2,7]+[2,9]+..., where [2,odd] indicates a choice, e.g., we may have 1+2, or 1+3+2, or 1+3+5+2+9, etc. Then A000700(n) is the number of elements of S that equal n. Also A000700(n) is the same parity as A000041(n) (the partition numbers). - _Jon Perry_, Dec 18 2003

%C a(n) is for n >= 2 the number of conjugacy classes of the symmetric group S_n which split into two classes under restriction to A_n, the alternating group. See the G. James - A. Kerber reference given under A115200, p. 12, 1.2.10 Lemma and the W. Lang link under A115198.

%C Also number of partitions of n such that if k is the largest part, then k occurs an odd number of times and each integer from 1 to k-1 occurs a positive even number of times (these are the conjugates of the partitions of n into distinct odd parts). Example: a(15)=4 because we have [3,3,3,2,2,1,1], [3,2,2,2,2,1,1,1,1], [3,2,2,1,1,1,1,1,1,1,1] and [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]. - _Emeric Deutsch_, Apr 16 2006

%C The INVERTi transform of A000009 (number of partitions of n into odd parts starting with offset 1) = (1, 0, 1, -1, 1, -1, 1, -2, 2, -2, 2, -3, 3, -3, 4, ...); = left border of triangle A146061. - _Gary W. Adamson_, Oct 26 2008

%C For n even: the sum over all even nonnegative integers, k, such that k^2 < n, of the number of partitions of (n-k^2)/2 into parts of size at most k. For n odd: the sum over all odd nonnegative integers, j, such that j^2 < n, of the number of partitions of (n-j^2)/2 into parts of size at most j. - _Graham H. Hawkes_, Oct 18 2013

%C This number is also (the number of conjugacy classes of S_n containing even permutations) - (the number of conjugacy classes of S_n containing odd permutations) = (the number of partitions of n into a number of parts having the same parity as n) - (the number of partitions of n into a number of parts having opposite parity as n) = (the number of partitions of n with largest part having same parity as n) - (the number of partitions with largest part having opposite parity as n). - _David L. Harden_, Dec 09 2016

%C a(n) is odd iff n belongs to A052002; that is, Sum_{n>=0} x^A052002(n) == Sum_{n>=0} a(n)*x^n (mod 2). - _Peter Bala_, Jan 22 2017

%C Also the number of conjugacy classes of S_n whose members yield unique square roots, i.e., there exists a unique h in S_n such that hh = g for any g in such a conjugacy class. Proof: first note that a permutation's square roots are determined by the product of the square roots of its decomposition into cycles of different lengths. h can only travel to one other cycle before it must "return home" (h^2(x) = g(x) must be in x's cycle), and, because if g^n(x) = x then h^2n(x) = x and h^2n(h(x)) = h(x), this "traveling" must preserve cycle length or one cycle will outpace the other. However, a permutation decomposing into two cycles of the same length has multiple square roots: for example, e = e^2 = (a b)^2, (a b)(c d) = (a c b d)^2 = (a d b c)^2, (a b c)(d e f) = (a d b e c f)^2 = (a e b f c d)^2, etc. This is true for any cycle length so we need only consider permutations with distinct cycle lengths. Finally, even cycle lengths are odd permutations and thus cannot be square, while odd cycle lengths have the unique square root h(x) = g^((n+1)/2)(x). Thus there is a correspondence between these conjugacy classes and partitions into distinct odd parts. - _Keith J. Bauer_, Jan 09 2024

%D R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; see p. 197.

%D B. C. Berndt, Ramanujan's theory of theta-functions, Theta functions: from the classical to the modern, Amer. Math. Soc., Providence, RI, 1993, pp. 1-63. MR 94m:11054.

%D T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, see q_2.

%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 277, Theorems 345, 347.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%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="/A000700/b000700.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from T. D. Noe)

%H K. Alladi, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00193-9">A variation on a theme of Sylvester - a smoother road to Gollnitz (Big) theorem</a>, Discrete Math., 196 (1999), 1-11.

%H Cristina Ballantine, Hannah E. Burson, Amanda Folsom, Chi-Yun Hsu, Isabella Negrini and Boya Wen, <a href="https://arxiv.org/abs/2109.00609">On a Partition Identity of Lehmer</a>, arXiv:2109.00609 [math.CO], 2021.

%H J. Dousse, <a href="https://doi.org/10.1090/proc/13376">Siladic's theorem: Weighted words, refinement and companion</a>, Proceedings of the American Mathematical Society, 145 (2017), 1997-2009.

%H J. A. Ewell, <a href="http://dx.doi.org/10.1155/S0161171200003902">Recursive determination of the enumerator for sums of three squares</a>, Internat. J. Math. and Math. Sci, 24 (2000), 529-532.

%H E. Friedman, <a href="/A000700/a000700.gif">Illustration of initial terms</a>.

%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 Edray Herber Goins and Talitha M. Washington, <a href="https://arxiv.org/abs/0909.5459">On the generalized climbing stairs problem</a>, Ars Combin. 117 (2014), 183-190. MR3243840 (Reviewed), arXiv:0909.5459 [math.CO], 2009.

%H H. Gupta, <a href="https://doi.org/10.1016/0097-3165(76)90051-0">Combinatorial proof of a theorem on partitions into an even or odd number of parts</a>, J. Combinatorial Theory Ser. A 21 (1976), no. 1, 100-103.

%H R. K. Guy, <a href="/A000700/a000700.pdf">A theorem in partitions</a>, Research Paper 11, Jan. 1967, Math. Dept., Univ. of Calgary. [Annotated scanned copy]

%H Christopher R. H. Hanusa and Rishi Nath, <a href="http://arxiv.org/abs/1201.6629">The number of self-conjugate core partitions</a>, arxiv:1201.6629 [math.NT], 2012.

%H Christian Kassel and Christophe Reutenauer, <a href="https://arxiv.org/abs/1505.07229v3">The zeta function of the Hilbert scheme of n points on a two-dimensional torus</a>, arXiv:1505.07229v3 [math.AG], 2015. [A later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication below.]

%H Christian Kassel and Christophe Reutenauer, <a href="https://arxiv.org/abs/1610.07793">Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus</a>, arXiv:1610.07793 [math.NT], 2016.

%H Martin Klazar, <a href="http://arxiv.org/abs/1808.08449">What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I</a>, arXiv:1808.08449 [math.CO], 2018.

%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 Mircea Merca, <a href="http://dx.doi.org/10.1016/j.jnt.2015.08.014">Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer</a>, Journal of Number Theory, Volume 160, March 2016, pp. 60-75, function p_s(n).

%H M. Osima, <a href="http://dx.doi.org/10.4153/CJM-1952-034-x">On the irreducible representations of the symmetric group</a>, Canad. J. Math., 4 (1952), 381-384.

%H Padmavathamma, R. Raghavendra and B. M. Chandrashekara, <a href="http://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 Igor Pak and Greta Panova, <a href="http://arxiv.org/abs/1304.5044">Unimodality via Kronecker products</a>, arXiv preprint arXiv:1304.5044 [math.CO], 2013.

%H Igor Pak and Greta Panova, <a href="https://doi.org/10.1016/j.laa.2020.05.005">Bounds on Kronecker coefficients via contingency tables</a>, Linear Algebra and its Applications (2020), Vol. 602, 157-178.

%H J. Perry, <a href="http://web.archive.org/web/20060923015305/http://www.users.globalnet.co.uk/~perry/maths/yetmorepartitionfunction/yetmorepartitionfunction.htm"> Yet More Partition Function</a>. [Archived copy as of Sep 23 2006 from web.archive.org]

%H N. Robbins, <a href="https://projecteuclid.org/euclid.rmjm/1181071694">Some identities connecting partition functions to other number theoretic functions</a>, Rocky Mountain J. Math. Volume 29, Number 1 (1999), 335-345.

%H I. Siladic, <a href="https://doi.org/10.3336/gm.52.1.05">Twisted SL(C,3)~- modules and combinatorial identities</a>, Glasnick Matematicki, 52 (2017), 53-77.

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>.

%H G. N. Watson, <a href="https://doi.org/10.1112/plms/s2-42.1.550">Two tables of partitions</a>, Proc. London Math. Soc., 42 (1936), 550-556.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Self-ConjugatePartition.html">Self-Conjugate Partition</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PartitionFunctionP.html">Partition Function P</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.

%H Mark Wildon, <a href="http://arxiv.org/abs/math/0609175">Counting Partitions on the Abacus</a>, arXiv:math/0609175 [math.CO], 2006.

%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>

%F G.f.: Product_{k>=1} (1 + x^(2*k-1)).

%F G.f.: Sum_{k>=0} x^(k^2)/Product_{i=1..k} (1-x^(2*i)). - Euler (Hardy and Wright, Theorem 345)

%F G.f.: 1/Product_{i>=1} (1 + (-x)^i). - _Jon Perry_, May 27 2004

%F Expansion of chi(q) = (-q; q^2)_oo = f(q) / f(-q^2) = phi(q) / f(q) = f(-q^2) / psi(-q) = phi(-q^2) / f(-q) = psi(q) / f(-q^4), where phi(), chi(), psi(), f() are Ramanujan theta functions.

%F Sum_{k=0..n} A081360(k)*a(n-k) = 0, for n > 0. - _John W. Layman_, Apr 26 2000

%F Euler transform of period-4 sequence [1, -1, 1, 0, ...].

%F Expansion of q^(1/24) * eta(q^2)^2 /(eta(q) * eta(q^4)) in powers of q. - _Michael Somos_, Jun 11 2004

%F Asymptotics: a(n) ~ exp(Pi*l_n)/(2*24^(1/4)*l_n^(3/2)) where l_n = (n-1/24)^(1/2) (Ayoub). The asymptotic formula in Ayoub is incorrect, as that would imply faster growth than the total number of partitions. (It was quoted correctly, the book is just wrong, not sure what the correct asymptotic is.) - _Edward Early_, Nov 15 2002. Right formula is a(n) ~ exp(Pi*sqrt(n/6)) / (2*24^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Jun 23 2014

%F a(n) = (1/n)*Sum_{k = 1..n} (-1)^(k+1)*b(k)*a(n-k), n>1, a(0) = 1, b(n) = A000593(n) = sum of odd divisors of n. - _Vladeta Jovovic_, Jan 19 2002 [see Theorem 2(a) in N. Robbins's article]

%F For n > 0: a(n) = b(n, 1) where b(n, k) = b(n-k, k+2) + b(n, k+2) if k < n, otherwise (n mod 2) * 0^(k-n). - _Reinhard Zumkeller_, Aug 26 2003

%F Expansion of q^(1/24) * (m * (1 - m) / 16)^(-1/24) in powers of q where m = k^2 is the parameter and q is the nome for Jacobian elliptic functions.

%F Given g.f. A(x), B(q) = (1/q)* A(q^3)^8 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u*v * (u - v^2) * (v - u^2) - (4 * (1 - u*v))^2. - _Michael Somos_, Jul 16 2007

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = f(t) where q = exp(2 Pi i t). - _Michael Somos_, Jul 16 2007

%F Expansion of q^(1/24)*f(t) in powers of q = exp(Pi*i*t) where f() is Weber's function. - _Michael Somos_, Oct 18 2007

%F A069911(n) = a(2*n + 1). A069910(n) = a(2*n).

%F a(n) = Sum_{k=1..n} (-1)^(n-k) A008284(n,k). - _Jeremy L. Martin_, Jul 06 2013

%F a(n) = S(n,1), where S(n,m) = Sum_{k=m..n/2} (-1)^(k+1)*S(n-k,k) + (-1)^(n+1), S(n,n)=(-1)^(n+1), S(0,m)=1, S(n,m)=0 for n < m. - _Vladimir Kruchinin_, Sep 07 2014

%F G.f.: Product_{k>0} (1 + x^(2*k-1)) = Product_{k>0} (1 - (-x)^k) / (1 - (-x)^(2*k)) = Product_{k>0} 1 / (1 + (-x)^k). - _Michael Somos_, Nov 08 2014

%F a(n) ~ Pi * BesselI(1, Pi*sqrt(24*n-1)/12) / sqrt(24*n-1) ~ exp(Pi*sqrt(n/6)) / (2^(7/4) * 3^(1/4) * n^(3/4)) * (1 - (3*sqrt(6)/(8*Pi) + Pi/(48*sqrt(6))) / sqrt(n) + (5/128 - 45/(64*Pi^2) + Pi^2/27648) / n). - _Vaclav Kotesovec_, Jan 08 2017

%F G.f.: exp(Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - _Ilya Gutkovskiy_, Jun 07 2018

%F Given g.f. A(x), B(q) = (1/q) * A(q^24) / 2^(1/4) satisfies 0 = f(B(q), B(q^5)) where f(u, v) = u^6 + v^6 + 2*u*v * (1 - (u*v)^4). - _Michael Somos_, Mar 14 2019

%F G.f.: Sum_{n >= 0} x^n/Product_{i = 1..n} ( 1 + (-1)^(i+1)*x^i ). - _Peter Bala_, Nov 30 2020

%F From _Peter Bala_, Jan 15 2021: (Start)

%F G.f.: (1 + x) * Sum_{n >= 0} x^(n*(n+2))/Product_{k = 1..n} (1 - x^(2*k)) = (1 + x)*(1 + x^3) * Sum_{n >= 0} x^(n*(n+4))/Product_{k = 1..n} (1 - x^(2*k)) = (1 + x)*(1 + x^3)*(1 + x^5) * Sum_{n >= 0} x^(n*(n+6))/ Product_{k = 1..n} (1 - x^(2*k)) = ....

%F G.f.: 1/(1 + x) * Sum_{n >= 0} x^(n-1)^2/Product_{k = 1..n} (1 - x^(2*k)) = 1/((1 + x)*(1 + x^3)) * Sum_{n >= 0} x^(n-2)^2/Product_{k = 1..n} (1 - x^(2*k)) = 1/((1 + x)*(1 + x^3)*(1 + x^5)) * Sum_{n >= 0} x^(n-3)^2/ Product_{k = 1..n} (1 - x^(2*k)) = .... (End)

%F a(n) = A046682(n) - A000701(n). See Gupta and also Ballantine et al. - _Michel Marcus_, Sep 04 2021

%F G.f.: A(x) = exp( Sum_{k >= 1} (-1)^k/(k*(x^k - x^(-k))) ). - _Peter Bala_, Dec 23 2021

%e T96a = 1/q + q^23 + q^71 + q^95 + q^119 + q^143 + q^167 + 2*q^191 + ...

%e G.f. = 1 + x + x^3 + x^4 + x^5 + x^6 + x^7 + 2*x^8 + 2*x^9 + 2*x^10 + 2*x^11 + 3*x^12 + ...

%p N := 100; t1 := series(mul(1+x^(2*k+1),k=0..N),x,N); A000700 := proc(n) coeff(t1,x,n); end;

%p # second Maple program:

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(n>i^2, 0,

%p b(n, i-1)+`if`(i*2-1>n, 0, b(n-(i*2-1), i-1))))

%p end:

%p a:= n-> b(n, iquo(n+1, 2)):

%p seq(a(n), n=0..80); # _Alois P. Heinz_, Mar 12 2016

%t CoefficientList[ Series[ Product[1 + x^(2k + 1), {k, 0, 75}], {x, 0, 70}], x] (* _Robert G. Wilson v_, Aug 22 2004 *)

%t a[ n_] := With[ {m = InverseEllipticNomeQ[ q]}, SeriesCoefficient[ ((1 - m) m /(16 q))^(-1/24), {q, 0, n}]]; (* _Michael Somos_, Jul 11 2011 *)

%t a[ n_] := SeriesCoefficient[ Product[1 + x^k, {k, 1, n, 2}], {x, 0, n}]; (* _Michael Somos_, Jul 11 2011 *)

%t p[n_] := p[n] = Select[Select[IntegerPartitions[n], DeleteDuplicates[#] == # &], Apply[And, OddQ[#]] &]; Table[p[n], {n, 0, 20}] (* shows partitions of n into distinct odd parts *)

%t Table[Length[p[n]], {n, 0, 20}] (* A000700(n), n >= 0 *)

%t conjugatePartition[part_] := Table[Count[#, _?(# >= i &)], {i, First[#]}] &[part]; s[n_] := s[n] = Select[IntegerPartitions[n], conjugatePartition[#] == # &]; Table[s[n], {n, 1, 20}] (* shows self-conjugate partitions *)

%t Table[Length[s[n]], {n, 1, 20}] (* A000700(n), n >= 1 *)

%t (* _Peter J. C. Moses_, Mar 12 2014 *)

%t CoefficientList[QPochhammer[q^2]^2/(QPochhammer[q]*QPochhammer[q^4]) + O[q]^70, q] (* _Jean-François Alcover_, Nov 05 2015, after _Michael Somos_ *)

%t (O[x]^70 + 2/QPochhammer[-1, -x])[[3]] (* _Vladimir Reshetnikov_, Nov 20 2015 *)

%t nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[If[OddQ[k], poly[[j + 1]] += poly[[j - k + 1]]], {j, nmax, k, -1}];, {k, 2, nmax}]; poly (* _Vaclav Kotesovec_, Nov 24 2017 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A)), n))}; /* _Michael Somos_, Jun 11 2004 */

%o (PARI) {a(n) = if( n<0, 0, polcoeff( 1 / prod( k=1, n, 1 + (-x)^k, 1 + x * O(x^n)), n))}; /* _Michael Somos_, Jun 11 2004 */

%o (PARI) my(x='x+O('x^70)); Vec(eta(x^2)^2/(eta(x)*eta(x^4))) \\ _Joerg Arndt_, Sep 07 2023

%o (Maxima)

%o S(n,m):=if n=0 then 1 else if n<m then 0 else if n=m then (-1)^(n+1) else sum((-1)^(k+1)*S(n-k,k),k,m,n/2)+(-1)^(n+1);

%o makelist(S(n,1),n,0,27); /* _Vladimir Kruchinin_, Sep 07 2014 */

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def A000700(n): return 1 if n== 0 else sum((-1)**(k+1)*A000700(n-k)*prod((p**(e+1)-1)//(p-1) for p, e in factorint(k).items() if p > 2) for k in range(1,n+1))//n # _Chai Wah Wu_, Sep 09 2021

%o (Magma)

%o m:=80;

%o R<x>:=PowerSeriesRing(Integers(), m);

%o Coefficients(R!( (&*[1 + x^(2*j+1): j in [0..m+2]]) )); // _G. C. Greubel_, Sep 07 2023

%o (SageMath)

%o from sage.modular.etaproducts import qexp_eta

%o m=80

%o def f(x): return qexp_eta(QQ[['q']], m+2).subs(q=x)

%o def A000700_list(prec):

%o P.<x> = PowerSeriesRing(QQ, prec)

%o return P( f(x^2)^2/(f(x)*f(x^4)) ).list()

%o A000700_list(m) # _G. C. Greubel_, Sep 07 2023

%Y Cf. A000009, A000041, A000701, A046682, A052002, A053250, A069910, A069911, A081362 (a signed version), A085547, A088994 (labeled version), A146061, A169987 - A169995, A295291, A304044.

%Y Main diagonal of A218907.

%K nonn,easy,nice

%O 0,9

%A _N. J. A. Sloane_