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JavaScript (Node.js), 36 bytes f=a=>a?.map?[a.length,...f(a[0])]:[] Try it online!
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# [JavaScript (Node.js)], 36 bytes <!-- language-all: lang-javascript --> f=a=>a?.map?[a.length,...f(a[0])]:[] [Try it online!][TIO-khtyawxl] [JavaScript (Node.js)]: https://nodejs.org [TIO-khtyawxl]: https://tio.run/##dY9BS8MwFIDv/RXPi01qTSoOhY26m@BFDwoe0h4ebbp2ZElI0rmB/72mDC9iTy9833sfZI9H9I0bbLjVppXTqcTyiWdTN0/csgParUCmpN6FPmeMdQRFUdN6LeqJc/h4eYNmdE7qoM4weunBqFY6OErnB6PBdPAaw2zvk7j@KcFiaHroIBjwo7XGBQi9BGNDXEcFTY@DHvQuEukwGAdk9gqD9OG3BaicxPb8m/AwBJpkHDcdlKDlFzyPupmLJHUyjE5DevMeXOySE2XeqiEQXvEq@66yinMq7moW9YFQ5qRV2EiSXr6f5inBq7I8mqGF4np@6lGpLc5yfaE0pZTQTZJsGqO9UZIpsyMdETWN9C8s/qWiXsD5klnlqyiL/H7Bi1k/1EtWPM7X9cVPPw "JavaScript (Node.js) – Try It Online"