A198954 Expansion of the rotational partition function for a heteronuclear diatomic molecule.
1, 3, 0, 5, 0, 0, 7, 0, 0, 0, 9, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 27, 0, 0, 0, 0, 0, 0, 0
Offset: 0
Examples
G.f. = 1 + 3*x + 5*x^3 + 7*x^6 + 9*x^10 + 11*x^15 + 13*x^21 + 15*x^28 + ... G.f. = 1 + 3*q^2 + 5*q^6 + 7*q^12 + 9*q^20 + 11*q^30 + 13*q^42 + 15*q^56 + ... Triangle begins: 1; 3, 0; 5, 0, 0; 7, 0, 0, 0; 9, 0, 0, 0, 0; 11, 0, 0, 0, 0, 0; 13, 0, 0, 0, 0, 0, 0; 15, 0, 0, 0, 0, 0, 0, 0; 17, 0, 0, 0, 0, 0, 0, 0, 0;
References
- G. H. Wannier, Statistical Physics, Dover Publications, 1987, see p. 215 equ. (11.13).
Links
- Antti Karttunen, Table of n, a(n) for n = 0..10011
Programs
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Maple
seq(op([2*i+1,0$i]), i=0..10); # Robert Israel, Jan 15 2015
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Mathematica
a[ n_] := If[ n < 0, 0, With[ {m = Sqrt[8 n + 1]}, If[ IntegerQ[m], m KroneckerSymbol[ 4, m], 0]]]; (* Michael Somos, Aug 26 2015 *) -
PARI
{a(n) = my(m); if( issquare( 8*n + 1, &m), m, 0)};
Formula
G.f.: Sum_{k>=0} (2*k + 1) * x^( (k^2 + k) / 2). This is related to Jacobi theta functions.
a(n) = (t*(t+1)-2*n-1)*(t-r), where t = floor(sqrt(2*(n+1))+1/2) and r = floor(sqrt(2*n)+1/2). - Mikael Aaltonen, Jan 15 2015
a(n) = abs(A010816(n)). - Joerg Arndt, Jan 16 2015
Comments