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 A198954 Expansion of the rotational partition function for a heteronuclear diatomic molecule. 3
 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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Last modified May 18 23:58 EDT 2021. Contains 344009 sequences. (Running on oeis4.)