Search for a doubly charged <math><mi>D</mi><mi>D</mi><mi>K</mi></math> bound state in <math><mi>ϒ</mi><mo>(</mo><mn>1</mn><mi>S</mi><mo>,</mo><mtext> </mtext><mn>2</mn><mi>S</mi><mo>)</mo></math> inclusive decays and via direct production in <math><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> collisions at <math><msqrt><mi>s</mi></msqrt><mo>=</mo><mn>10.520</mn></math>, 10.580, and 10.867 GeV

We report the results of a first search for a doubly charged DDK bound state, denoted the R++, in ϒ(1S) and ϒ(2S) inclusive decays and via direct production in e+e− collisions at s=10.520, 10.580, and 10.867 GeV. The search uses data accumulated with the Belle detector at the KEKB asymmetric-energy e+e− collider. No significant signals are observed in the D+Ds*+ invariant-mass spectra of all studied modes. The 90% credibility level upper limits on their product branching fractions in ϒ(1S) and ϒ(2S) inclusive decays (B(ϒ(1S, 2S)→R+++anything)&#215;B(R++→D+Ds*+)), the product values of Born cross section and branching fraction in e+e− collisions (σ(e+e−→R+++anything)&#215;B(R++→D+Ds*+)) at s=10.520, 10.580, and 10.867 GeV under different assumptions of R++ masses varying from 4.13 to 4.17 GeV/c2 and widths varying from 0 to 5 MeV are obtained.