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%I #44 Jan 22 2020 20:12:14
%S 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,
%T 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,
%U 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
%N Expansion of the rotational partition function for a heteronuclear diatomic molecule.
%C 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.
%C 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
%C 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
%D G. H. Wannier, Statistical Physics, Dover Publications, 1987, see p. 215 equ. (11.13).
%H Antti Karttunen, <a href="/A198954/b198954.txt">Table of n, a(n) for n = 0..10011</a>
%F G.f.: Sum_{k>=0} (2*k + 1) * x^( (k^2 + k) / 2). This is related to Jacobi theta functions.
%F 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
%F a(n) = A053187(2n+1) - A053187(2n). - _Robert Israel_, Jan 15 2015
%F a(n) = abs(A010816(n)). - _Joerg Arndt_, Jan 16 2015
%e G.f. = 1 + 3*x + 5*x^3 + 7*x^6 + 9*x^10 + 11*x^15 + 13*x^21 + 15*x^28 + ...
%e G.f. = 1 + 3*q^2 + 5*q^6 + 7*q^12 + 9*q^20 + 11*q^30 + 13*q^42 + 15*q^56 + ...
%e Triangle begins:
%e 1;
%e 3, 0;
%e 5, 0, 0;
%e 7, 0, 0, 0;
%e 9, 0, 0, 0, 0;
%e 11, 0, 0, 0, 0, 0;
%e 13, 0, 0, 0, 0, 0, 0;
%e 15, 0, 0, 0, 0, 0, 0, 0;
%e 17, 0, 0, 0, 0, 0, 0, 0, 0;
%p seq(op([2*i+1,0$i]), i=0..10); # _Robert Israel_, Jan 15 2015
%t 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 *)
%o (PARI) {a(n) = my(m); if( issquare( 8*n + 1, &m), m, 0)};
%Y Cf. A053187, A107270.
%K nonn,tabl,easy
%O 0,2
%A _Michael Somos_, Oct 31 2011
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