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R, 126 114 108107 bytes

\(v,S=\`^`=\(n,k)`if`(n<1,k<1,k*Sk*(n-1,k)+S^k+(n-1,)^(k-1))*(k>=0))lapply(l<-(seq(v)-1),\(k)sum(v*sapply(l,\(n)S(n,k)n^k)))

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R, 126 114 108 bytes

\(v,S=\(n,k)`if`(n<1,k<1,k*S(n-1,k)+S(n-1,k-1))*(k>=0))lapply(l<-(seq(v)-1),\(k)sum(v*sapply(l,\(n)S(n,k))))

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R, 126 114 107 bytes

\(v,`^`=\(n,k)`if`(n<1,k<1,k*(n-1)^k+(n-1)^(k-1))*(k>=0))lapply(l<-(seq(v)-1),\(k)sum(v*sapply(l,\(n)n^k)))

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R, 126 114114 108 bytes

\(v,S=\(n,k)(n>=0&k>=0)*`if``if`(n<1,+k<1k<1,k*S(n-1,k)+S(n-1,k-1))*(k>=0))lapply(l<-(seq(v)-1),\(k)sum(v*sapply(l,\(n)S(n,k))))

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R, 126 114 bytes

\(v,S=\(n,k)(n>=0&k>=0)*`if`(n<1,+k<1,k*S(n-1,k)+S(n-1,k-1)))lapply(l<-(seq(v)-1),\(k)sum(v*sapply(l,\(n)S(n,k))))

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R, 126 114 108 bytes

\(v,S=\(n,k)`if`(n<1,k<1,k*S(n-1,k)+S(n-1,k-1))*(k>=0))lapply(l<-(seq(v)-1),\(k)sum(v*sapply(l,\(n)S(n,k))))

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R, 126126 114 bytes

\(v){,S=\(n,k)`if`(n<0||k<0,0,`if`n>=0&k>=0)*`if`(n<1,+k<1,k*S(n-1,k)+S(n-1,k-1)))
n=sumlapply(v|1l<-(seq(v)-1
sapply(0:n),\(k)sum(v*sapply(0:nl,\(n)S(n,k))))}

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R, 126 bytes

\(v){S=\(n,k)`if`(n<0||k<0,0,`if`(n<1,+k<1,k*S(n-1,k)+S(n-1,k-1)))
n=sum(v|1)-1
sapply(0:n,\(k)sum(v*sapply(0:n,\(n)S(n,k))))}

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R, 126 114 bytes

\(v,S=\(n,k)(n>=0&k>=0)*`if`(n<1,+k<1,k*S(n-1,k)+S(n-1,k-1)))lapply(l<-(seq(v)-1),\(k)sum(v*sapply(l,\(n)S(n,k))))

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