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A000700
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Expansion of product (1+x^(2k+1)), k=0..inf; 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)
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834
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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
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OFFSET
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0,9
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COMMENTS
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Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
McKay-Thompson series of class 96a for the Monster group with a(0) = 0.
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]
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REFERENCES
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K. Alladi, A variation on a theme of Sylvester - a smoother road to Gollnitz (Big) theorem, Discrete Math., 196 (1999), 1-11.
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.
J. A. Ewell, Recursive determination of the enumerator for sums of three squares, Internat. J. Math. and Math. Sci, 24 (2000), 529-532.
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.
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.
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).
G. N. Watson, Two tables of partitions, Proc. London Math. Soc., 42 (1936), 550-556.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..1000
E. Friedman, Illustration of initial terms
Christopher R. H. Hanusa and Rishi Nath, The number of self-conjugate core partitions (2012), arxiv:1201.6629.
J. Perry, Yet More Partition Function [Archived copy as of Sep 23 2006 from web.archive.org]
M. Somos, Introduction to Ramanujan theta functions
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
Index entries for McKay-Thompson series for Monster simple group
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FORMULA
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G.f.: prod(k>=1, 1+x^(2*k-1)) = sum(k>=0, x^(k^2)/prod(i=1..k, 1-x^(2*i) )). - Euler (Hardy and Wright, Theorem 345).
G.f.: 1/prod(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[b(k)*a(n-k), k=0..n]=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 (efedula(AT)math.mit.edu), Nov 15 2002
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
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(x) = (1/x)* A(x^3)^8 satisfies 0 = f(B(x), B(x^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 Fourier series of a weight 0 level 2304 modular form. 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).
G.f.: G(0)/x where G(k) = 1 - 1/(1 - 1/(1 - 1/(1+(x)^(2*k+1))/G(k+1) )); (continued fraction, see A006950). - Sergei N. Gladkovskii, Jan 28 2013
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EXAMPLE
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T96a = 1/q + q^23 + q^71 + q^95 + q^119 + q^143 + q^167 + 2q^191 + ...
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 + ...
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MAPLE
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N := 100; t1 := series(mul(1+x^(2*k+1), k=0..N), x, N); A000700 := proc(n) coeff(t1, x, n); end;
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MATHEMATICA
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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 *)
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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)^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 */
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CROSSREFS
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Cf. A000009, A000041, A000701, A046682, A085547, A053250, A081362 (a signed version), A169987-A169995, A069910, A069911, A146061.
Sequence in context: A169995 A213419 * A081362 A225956 A112216 A058688
Adjacent sequences: A000697 A000698 A000699 * A000701 A000702 A000703
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Links fixed by M. F. Hasler, Jan 06 2013
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STATUS
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approved
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