0,1

Ramanujan's theta function f(a,b)=Sum a^{n*(n+1)/2}*b^{n*(n-1)/2}, n=-inf..inf.

Euler transform of period 2 sequence [1,-1,...].

This sequence is the concatenation of the base-b digits in the sequence b^n, for any base b >= 2. - Davis Herring (herring(AT)lanl.gov), Nov 16 2004

Number of partitions of n into distinct parts such that the greatest part equals the number of all parts, see also A047993; a(n)=A117195(n,0) for n>0; a(n)=1-A117195(n,1) for n>1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 03 2006

Triangle T(n,k), 0<=k<=n, read by rows, given by A000007 DELTA A000004 where DELTA is the operator defined in A084938 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 03 2009]

Convolved with A000041 = A022567, the convolution square of A000009 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 11 2009]

Index entries for characteristic functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

G.f.: theta2(q)/(2*q^(1/4)) = f(q, q^3) where f is Ramanujan's theta function.

G.f.: Product_{k>0} (1-x^(2*k))/(1-x^(2*k-1)). - Vladeta Jovovic (vladeta(AT)eunet.rs), May 02 2002

a(0)=1; for n>0, a(n)=A002024(n+1)-A002024(n). - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 05 2004

G.f.: sum(j=0, oo, product(k=0, j, x^j)) - Jon Perry (perry(AT)globalnet.co.uk), Mar 30 2004

Expansion of q^(-1/8)eta(q^2)^2/eta(q) in powers of q.

Given g.f. A(x), then B(x)=x*A(x^8) satisfies 0=f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6)=u1*u6^3 +u2*u3^3 -u1*u2^2*u6. - Michael Somos Apr 13 2005

a(n)=b(8n+1) where b(n) is multiplicative and b(2^e)=0^e, b(p^e)=(1+(-1)^e)/2 if p>2. - Michael Somos Jun 06 2005

a(n) = floor((1-cos(Pi*sqrt(8*n+1)))/2) - Carl R. White (oeisfan(AT)cyreksoft.yorks.com), Mar 18 2006

a(n)=round(sqrt(2n+1))-round(sqrt(2n)). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 06 2007

a(n)=ceiling(2*sqrt(2n+1))-floor(2*sqrt(2n))-1. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 06 2007

a(n) = f(n,0) with f(x,y) = if x>0 then f(x-y,y+1) else 0^(-x). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 27 2008]

Comment from Philippe DELEHAM, Jan 04 2008: As a triangle this begins:

.1;

.1, 0;

.1, 0, 0;

.1, 0, 0, 0;

.1, 0, 0, 0, 0;

.1, 0, 0, 0, 0, 0 ; ...

(PARI) a(n)=local(X); if(n<0, 0, X=x+x*O(x^n); polcoeff(eta(X^2)^2/eta(X), n))

(PARI) a(n)=if(n<0, 0, issquare(8*n+1))

Cf. A000217, A023531.

a(n) = A035214(n) - 1. Also a(n) = A005369(2n).

A022567 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 11 2009]

Adjacent sequences: A010051 A010052 A010053 this_sequence A010055 A010056 A010057

Sequence in context: A113430 A113681 A155972 this_sequence A106459 A143433 A143434

nonn,tabl

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

Additional comments from Michael Somos, Apr 27, 2000.