0,9

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 inequlatity if a part is even. [Alladi]

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

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 (perry(AT)globalnet.co.uk), Dec 18 2003

Normalized McKay-Thompson series of class 96a for the Monster group.

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 (deutsch(AT)duke.poly.edu), 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. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 26 2008]

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.

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.

T. D. Noe, Table of n, a(n) for n=0..1000

Index entries for McKay-Thompson series for Monster simple group

E. Friedman, Illustration of initial terms

J. Perry, Title?

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

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

Let b(n)=A081360(n); then Sum[b(k)*a(n-k), k=0..n]=0, for n>0 - John W. Layman (layman(AT)math.vt.edu), 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).

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 (vladeta(AT)eunet.rs), 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 (reinhard.zumkeller(AT)gmail.com), Aug 26 2003

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

G.f.: 1/prod(i=1, oo, 1+(-1)^i*x^i) - Jon Perry (perry(AT)globalnet.co.uk), May 27 2004

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

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 + ...

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

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

(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 */

Cf. A000009, A000041, A000701, A046682, A085547, A053250, A081362 (a signed version).

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

Cf. A146061 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 26 2008]

Sequence in context: A005854 A035435 A025775 this_sequence A081362 A112216 A058688

Adjacent sequences: A000697 A000698 A000699 this_sequence A000701 A000702 A000703

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).