cp's OEIS Frontend

A015664 Expansion of e.g.f. theta_3^(1/2).

%I A015664 #29 Jul 07 2023 14:57:19
%S A015664 1,1,-1,3,9,-15,135,-2205,21105,76545,694575,-6392925,-56600775,
%T A015664 66891825,-19964169225,741313447875,5375639894625,44667168170625,
%U A015664 -2328500019470625,5663134786183875,-466442955127524375,11513119609487120625
%N A015664 Expansion of e.g.f. theta_3^(1/2).
%C A015664 The sequence shows the coefficients of sqrt(theta_3) regarded as an exponential generating function.
%D A015664 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
%H A015664 Seiichi Manyama, <a href="/A015664/b015664.txt">Table of n, a(n) for n = 0..449</a>
%F A015664 E.g.f. appears to equal exp( Sum_{n >= 0} x^(2*n+1)/((2*n+1)*(1 + x^(2*n+1))) ). - _Peter Bala_, Dec 23 2021
%F A015664 a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} A186690(k) * a(n-k)/(n-k)!. - _Seiichi Manyama_, Jul 07 2023
%e A015664 sqrt(theta_3) = 1 + q - (1/2)*q^2 + (1/2)*q^3 + (3/8)*q^4 - (1/8)*q^5 + (3/16)*q^6 - (7/16)*q^7 + (67/128)*q^8 + (27/128)*q^9 + ...
%p A015664 # get basic theta series in maple
%p A015664 maxd:=201:
%p A015664 # get th2, th3, th4 = Jacobi theta constants out to degree maxd
%p A015664 temp0:=trunc(evalf(sqrt(maxd)))+2:
%p A015664 a:=0: for i from -temp0 to temp0 do a:=a+q^( (i+1/2)^2): od:
%p A015664 th2:=series(a,q,maxd); # A098108
%p A015664 a:=0: for i from -temp0 to temp0 do a:=a+q^(i^2): od:
%p A015664 th3:=series(a,q,maxd); # A000122
%p A015664 th4:=series(subs(q=-q,th3),q,maxd); # A002448
%p A015664 series(sqrt(th3),q,maxd); # this sequence
%t A015664 nmax = 25; CoefficientList[Series[EllipticTheta[3, 0, x]^(1/2), {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Oct 23 2018 *)
%Y A015664 Cf. A000122 (theta_3), A015680, A186690.
%Y A015664 Cf. A015665, A015666, A015667, A015669, A015671, A015672, A015673, A015675, A015676, A015677, A015678, A015679.
%K A015664 sign
%O A015664 0,4
%A A015664 _N. J. A. Sloane_
%E A015664 Entry revised by _N. J. A. Sloane_, Oct 22 2018