Summation of Catalan Convolution

an someone help me solve this.
$ \sum _{k=0}^{\infty } \frac{2^{-2-2 k} \binom{3+2 k}{k}}{4+k}$
Wolfram alpha says it converges to 1 , but actually I have no idea, how to solve it.

  • The number 4 in the numerator sounds a little suspicious. Could you please double check and make sure that there is no typo?

  • It is just multiplying, I will change it to be less vague.

  • This looks much better.

  • This is unlike any sequence I have seen before. How did you come up with this problem? Any motivation behind it?

  • I think I know how to get a solution. However, it already took me good half an hour just to develop a strategy, and it will take at least as long as that to properly write it down. The bounty is way too low for that, this question is in the 30$ range at least.

  • If that is a perk, the solution I have is almost purely combinatorial, very little hard computations involved.

  • Just checking if you read my comment? If you can't increase the bounty, I can give a very rough sketch of the solution instead, referencing all the material I need.

  • I went for the sketch. I won't be giving more details on the referenced sources (they prove everything though), because of the low bounty.

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