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library(shiny)
library(DT)

shinyApp(
  ui <- fluidPage(
    DT::dataTableOutput("data")
  ),

  server <- function(input, output) {

    shinyInput <- function(FUN, len, id, ...) {
      inputs <- character(len)
      for (i in seq_len(len)) {
        inputs[i] <- as.character(FUN(paste0(id, i), ...))
      }
      inputs
    }

    df <- reactiveValues(data = data.frame(
      Delete = shinyInput(actionButton, 10, 'button_', label = "Delete", onclick = 'Shiny.onInputChange(\"select_button\",  this.id)' ),
      Value1 = 1:10,
      Value2 = c("A", "B", "C", "D", "E"),
      stringsAsFactors = FALSE,
      row.names = 1:10
    ))

    output$data <- DT::renderDataTable(
      df$data, server = FALSE, escape = FALSE, selection = 'none'
    )

    observeEvent(input$select_button, {
      selectedRow <- as.numeric(strsplit(input$select_button, "_")[[1]][2])
      df$data <- df$data[rownames(df$data) != selectedRow, ]
    })

  }
)