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A000700 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.
(Formerly M0217 N0078)
1430
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, 20, 23, 25, 26, 29, 33, 35, 37, 41, 46, 49, 52, 57, 63, 68, 72, 78, 87, 93, 98, 107, 117, 125, 133, 144, 157, 168, 178, 192, 209, 223, 236, 255, 276, 294, 312, 335, 361, 385 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

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

Coefficients of replicable function number 96a. - N. J. A. Sloane, Jun 10 2015

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

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

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

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.

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

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

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

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

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

REFERENCES

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

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.

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

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.

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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)

K. Alladi, A variation on a theme of Sylvester - a smoother road to Gollnitz (Big) theorem, Discrete Math., 196 (1999), 1-11.

J. A. Ewell, Recursive determination of the enumerator for sums of three squares, Internat. J. Math. and Math. Sci, 24 (2000), 529-532.

E. Friedman, Illustration of initial terms

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).

Edray Herber Goins and Talitha M. Washington, On the generalized climbing stairs problem, Ars Combin.  117  (2014), 183-190.  MR3243840 (Reviewed), arXiv:0909.5459 [math.CO].

R. K. Guy, A theorem in partitions, Research Paper 11, Jan. 1967, Math. Dept., Univ. of Calgary [Annotated scanned copy]

Christopher R. H. Hanusa and Rishi Nath, The number of self-conjugate core partitions, arxiv:1201.6629 [math.NT], 2012.

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.

Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function p_s(n).

M. Osima, On the irreducible representations of the symmetric group, Canad. J. Math., 4 (1952), 381-384.

Padmavathamma, R. Raghavendra and B. M. Chandrashekara, A new bijective proof of a partition theorem of K. Alladi, Discrete Math., 237 (2004), 125-128.

I. Pak and G. Panova, Unimodality via Kronecker products, arXiv preprint arXiv:1304.5044 [math.CO], 2013.

J. Perry, Yet More Partition Function [Archived copy as of Sep 23 2006 from web.archive.org]

N. Robbins, Some identities connecting partition functions to other number theoretic functions, Rocky Mountain J. Math. Volume 29, Number 1 (1999), 335-345.

M. Somos, Introduction to Ramanujan theta functions

G. N. Watson, Two tables of partitions, Proc. London Math. Soc., 42 (1936), 550-556.

Eric Weisstein's World of Mathematics, Self-Conjugate Partition

Eric Weisstein's World of Mathematics, Partition Function P

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Mark Wildon, Counting Partitions on the Abacus, arXiv:math/0609175 [math.CO], 2006.

Index entries for McKay-Thompson series for Monster simple group

FORMULA

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

G.f.: 1/Product_{i>=1} (1+(-1)^i*x^i). - Jon Perry, May 27 2004

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.

Let b(n)=A081360(n); then Sum_{k=0..n} b(k)*a(n-k) = 0, for n>0. - John W. Layman, Apr 26 2000

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

Expansion of q^(1/24) * eta(q^2)^2 /(eta(q) * eta(q^4)) in powers of q. - Michael Somos, Jun 11 2004

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

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]

For n>0: a(n) = b(n, 1) where b(n, k) = if k<n then b(n-k, k+2) + b(n, k+2) else (n mod 2) * 0^(k-n). - Reinhard Zumkeller, Aug 26 2003

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.

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

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

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

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

a(n) = Sum_{k=1..n} (-1)^(n-k) A008284(n,k). - Jeremy L. Martin, Jul 06 2013

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

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

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

EXAMPLE

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

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 + ...

MAPLE

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

# second Maple program:

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

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

    end:

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

seq(a(n), n=0..80);  # Alois P. Heinz, Mar 12 2016

MATHEMATICA

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

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

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

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 *)

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

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 *)

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

(* Peter J. C. Moses, Mar 12 2014 *)

CoefficientList[QPochhammer[q^2]^2/(QPochhammer[q]*QPochhammer[q^4]) + O[q]^70, q] (* Jean-Fran├žois Alcover, Nov 05 2015, after Michael Somos *)

(O[x]^70 + 2/QPochhammer[-1, -x])[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

PROG

(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 */

(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 */

(Maxima)

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);

makelist(S(n, 1), n, 0, 27); /* Vladimir Kruchinin, Sep 07 2014 */

CROSSREFS

Cf. A000009, A000041, A000701, A046682, A085547, A053250, A081362 (a signed version), A169987-A169995, A069910, A069911, A146061.

Main diagonal of A218907. Cf. A052002.

Sequence in context: A169995 A213419 * A081362 A112216 A225956 A058688

Adjacent sequences:  A000697 A000698 A000699 * A000701 A000702 A000703

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

Gave a different combinatorial interpretation of the sequence (and two equivalents of it), in terms of a formula that has been already submitted. - David L. Harden, Dec 09 2016

STATUS

approved

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Last modified November 23 19:00 EST 2017. Contains 295128 sequences.