For problem statement at 1000-1999/1500-1599/1510-1519/1516/problemE.txt this is a correct solution, but verifier at 1000-1999/1500-1599/1510-1519/1516/verifierE.go ends with All tests passed can you fix the verifier? ```go
package main

import (
	"bufio"
	"fmt"
	"os"
)

const MOD = 1000000007

func main() {
	var n int64
	var k int
	reader := bufio.NewReader(os.Stdin)
	fmt.Fscan(reader, &n, &k)

	if k == 0 {
		return
	}

	limit := 2 * k

	// dp[x][m] stores the unsigned Stirling number of the first kind [x, x-m]
	// This corresponds to permutations of length x with x-m cycles.
	dp := make([][]int64, limit+1)
	for i := range dp {
		dp[i] = make([]int64, k+1)
	}

	dp[0][0] = 1
	for x := 1; x <= limit; x++ {
		dp[x][0] = 1
		for m := 1; m <= k; m++ {
			// Recurrence: [n, n-m] = [n-1, n-m-1] + (n-1)*[n-1, n-m]
			// Mapping to dp: dp[x][m] = dp[x-1][m] + (x-1)*dp[x-1][m-1]
			term1 := dp[x-1][m]
			term2 := (int64(x-1) * dp[x-1][m-1]) % MOD
			dp[x][m] = (term1 + term2) % MOD
		}
	}

	// Precompute factorials and inverse factorials for Lagrange Interpolation
	fact := make([]int64, limit+1)
	invFact := make([]int64, limit+1)
	fact[0] = 1
	for i := 1; i <= limit; i++ {
		fact[i] = (fact[i-1] * int64(i)) % MOD
	}
	invFact[limit] = power(fact[limit], int64(MOD-2))
	for i := limit - 1; i >= 0; i-- {
		invFact[i] = (invFact[i+1] * int64(i+1)) % MOD
	}

	// Calculate prefix and suffix products of (n - i)
	nMod := n % MOD
	preProd := make([]int64, limit+2)
	preProd[0] = 1
	for i := 0; i <= limit; i++ {
		val := (nMod - int64(i)) % MOD
		if val < 0 {
			val += MOD
		}
		preProd[i+1] = (preProd[i] * val) % MOD
	}

	sufProd := make([]int64, limit+2)
	sufProd[limit+1] = 1
	for i := limit; i >= 0; i-- {
		val := (nMod - int64(i)) % MOD
		if val < 0 {
			val += MOD
		}
		sufProd[i] = (sufProd[i+1] * val) % MOD
	}

	// Calculate Lagrange basis weights L_j
	L := make([]int64, limit+1)
	for j := 0; j <= limit; j++ {
		num := (preProd[j] * sufProd[j+1]) % MOD
		den := (invFact[j] * invFact[limit-j]) % MOD
		sign := int64(1)
		if (limit-j)%2 == 1 {
			sign = -1
		}
		term := (num * den) % MOD
		term = (term * sign) % MOD
		if term < 0 {
			term += MOD
		}
		L[j] = term
	}

	// Interpolate values for S(n, m)
	S_vals := make([]int64, k+1)
	for m := 0; m <= k; m++ {
		var val int64
		for j := 0; j <= limit; j++ {
			term := (dp[j][m] * L[j]) % MOD
			val = (val + term) % MOD
		}
		S_vals[m] = val
	}

	out := bufio.NewWriter(os.Stdout)
	defer out.Flush()

	// Compute final answers
	for i := 1; i <= k; i++ {
		var ans int64
		for j := i; j >= 0; j -= 2 {
			ans = (ans + S_vals[j]) % MOD
		}
		if i > 1 {
			out.WriteByte(' ')
		}
		fmt.Fprintf(out, "%d", ans)
	}
	out.WriteByte('\n')
}

func power(a, b int64) int64 {
	res := int64(1)
	a %= MOD
	for b > 0 {
		if b%2 == 1 {
			res = (res * a) % MOD
		}
		a = (a * a) % MOD
		b /= 2
	}
	return res
}
```