编程解决很简单,求和公式估计是没有的.

电脑解决就一句话搞定：
输入 Table[Sum[k!, {k, 1, n}], {n, 20}]
输出：
1, 3, 9, 33, 153, 873, 5913, 46233, 409113, 4037913, 43954713, \
522956313, 6749977113, 93928268313, 1401602636313, 22324392524313, \
378011820620313, 6780385526348313, 128425485935180313, \
2561327494111820313