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

0,2

Comments

The partition function of a heteronuclear diatomic molecule is Sum_{J>=0} (2*J + 1) * exp( - J * (J + 1) * hbar^2 / (2 * I * k * T)) where I is the moment of inertia, hbar is reduced Planck's constant, k is Boltzmann's constant, and T is temperature. The degeneracy for the J-th energy level is 2*J + 1.

As triangle: triangle T(n,k), read by rows, given by (3,-4/3,1/3,0,0,0,0,0,0,0,...) DELTA (0,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 01 2011

Note that the g.f. theta_1'(0, q^(1/2)) / (2 * q^(1/8)) = 1 - 3*q + 5*q^3 - 7*q^6 + 9*q^10 + ... which is the same as this sequence except the signs alternate. - Michael Somos, Aug 26 2015

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

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) = A053187(2n+1) - A053187(2n). - Robert Israel, Jan 15 2015

a(n) = abs(A010816(n)). - Joerg Arndt, Jan 16 2015

Example

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;

Maple

seq(op([2*i+1, 0$i]), i=0..10); # Robert Israel, Jan 15 2015

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 *)

Prog

(PARI) {a(n) = my(m); if( issquare( 8*n + 1, &m), m, 0)};

Crossrefs

Cf. A053187, A107270.

Sequence in context: A154725 A010816 A133089 * A136599 A227498 A131986

Adjacent sequences: A198951 A198952 A198953 * A198955 A198956 A198957

Keyword

nonn,tabl,easy

Author

Michael Somos, Oct 31 2011

Status

approved