This problem troubling me, I have some clue to the distribution of these pattern it includes the gamma function(upcourse) and the triangular numbers, but i can't make any successful formulation: $$a_1=0!1$$ $$a_2=1!2$$ $$a_3=2!3+0!2$$ $$a_4=3!4+1!3$$ $$a_5=4!5+2!4$$ $$a_6=5!6+3!5+0!3$$ $$a_7=6!7+4!6+1!4$$ $$a_8=7!8+5!7+2!5$$ $$a_9=8!9+6!8+3!6$$ $$a_{10}=9!10+7!9+4!7+0!4$$ $$a_{11}=10!11+8!10+5!8+1!5$$ $$a_{12}=11!12+9!11+6!9+2!6$$ $$a_{13}=12!13+10!12+7!10+3!7$$ $$a_{14}=13!14+11!13+8!11+4!8$$ $$a_{15}=14!15+12!14+9!12+5!9+0!5$$ $${.....}$$ is there a formula for this?
2026-03-30 13:35:26.1774877726
Is there any formula for this given pattern?
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