Use PMI to prove this identity











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Use Principle of Mathematical Induction to show that:



for $nge2$
$$sum_{{a_{n-1}}spacespace=1}^{a_n}sum_{a_{n-2}spacespace=1}^{a_{n-1}}......sum_{a_{1}=1}^{a_2}a_1=frac{prod_{i=0}^{n}(a_n+i)}{n!}$$



I have totally no idea to deal with the product. I can understand it when it only uses 3 sigma notation.






summation products







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  • I think one of the components of PMI lies on whether n=1 is satisfied, your lhs doesnt work on it , so no point in applying .For n>=2 this identity holds true ,I can prove it but not using PMI
    – swapedoc
    yesterday










  • @swapedoc How to prove it without PMI? Could you give me a hint? That may help.
    – yuanming luo
    yesterday












  • hint: look for faulhaber formula , also try to guess coefficients of x^k in (x+1)(x+2)(x+3)....(x+n) where k can be from 0 to n
    – swapedoc
    yesterday















up vote
0
down vote

favorite
1












Use Principle of Mathematical Induction to show that:



for $nge2$
$$sum_{{a_{n-1}}spacespace=1}^{a_n}sum_{a_{n-2}spacespace=1}^{a_{n-1}}......sum_{a_{1}=1}^{a_2}a_1=frac{prod_{i=0}^{n}(a_n+i)}{n!}$$



I have totally no idea to deal with the product. I can understand it when it only uses 3 sigma notation.






summation products







share|cite|improve this question









New contributor




yuanming luo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.




















  • any hint will be thankful
    – yuanming luo
    yesterday










  • Could anyone check whether it is correct?
    – yuanming luo
    yesterday










  • I think one of the components of PMI lies on whether n=1 is satisfied, your lhs doesnt work on it , so no point in applying .For n>=2 this identity holds true ,I can prove it but not using PMI
    – swapedoc
    yesterday










  • @swapedoc How to prove it without PMI? Could you give me a hint? That may help.
    – yuanming luo
    yesterday












  • hint: look for faulhaber formula , also try to guess coefficients of x^k in (x+1)(x+2)(x+3)....(x+n) where k can be from 0 to n
    – swapedoc
    yesterday













up vote
0
down vote

favorite
1









up vote
0
down vote

favorite
1






1





Use Principle of Mathematical Induction to show that:



for $nge2$
$$sum_{{a_{n-1}}spacespace=1}^{a_n}sum_{a_{n-2}spacespace=1}^{a_{n-1}}......sum_{a_{1}=1}^{a_2}a_1=frac{prod_{i=0}^{n}(a_n+i)}{n!}$$



I have totally no idea to deal with the product. I can understand it when it only uses 3 sigma notation.






summation products







share|cite|improve this question









New contributor




yuanming luo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











Use Principle of Mathematical Induction to show that:



for $nge2$
$$sum_{{a_{n-1}}spacespace=1}^{a_n}sum_{a_{n-2}spacespace=1}^{a_{n-1}}......sum_{a_{1}=1}^{a_2}a_1=frac{prod_{i=0}^{n}(a_n+i)}{n!}$$



I have totally no idea to deal with the product. I can understand it when it only uses 3 sigma notation.






summation products




summation products






share|cite|improve this question









New contributor




yuanming luo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question









New contributor




yuanming luo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









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yuanming luo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






yuanming luo is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.












  • any hint will be thankful
    – yuanming luo
    yesterday










  • Could anyone check whether it is correct?
    – yuanming luo
    yesterday










  • I think one of the components of PMI lies on whether n=1 is satisfied, your lhs doesnt work on it , so no point in applying .For n>=2 this identity holds true ,I can prove it but not using PMI
    – swapedoc
    yesterday










  • @swapedoc How to prove it without PMI? Could you give me a hint? That may help.
    – yuanming luo
    yesterday












  • hint: look for faulhaber formula , also try to guess coefficients of x^k in (x+1)(x+2)(x+3)....(x+n) where k can be from 0 to n
    – swapedoc
    yesterday


















  • any hint will be thankful
    – yuanming luo
    yesterday










  • Could anyone check whether it is correct?
    – yuanming luo
    yesterday










  • I think one of the components of PMI lies on whether n=1 is satisfied, your lhs doesnt work on it , so no point in applying .For n>=2 this identity holds true ,I can prove it but not using PMI
    – swapedoc
    yesterday










  • @swapedoc How to prove it without PMI? Could you give me a hint? That may help.
    – yuanming luo
    yesterday












  • hint: look for faulhaber formula , also try to guess coefficients of x^k in (x+1)(x+2)(x+3)....(x+n) where k can be from 0 to n
    – swapedoc
    yesterday
















any hint will be thankful
– yuanming luo
yesterday




any hint will be thankful
– yuanming luo
yesterday












Could anyone check whether it is correct?
– yuanming luo
yesterday




Could anyone check whether it is correct?
– yuanming luo
yesterday












I think one of the components of PMI lies on whether n=1 is satisfied, your lhs doesnt work on it , so no point in applying .For n>=2 this identity holds true ,I can prove it but not using PMI
– swapedoc
yesterday




I think one of the components of PMI lies on whether n=1 is satisfied, your lhs doesnt work on it , so no point in applying .For n>=2 this identity holds true ,I can prove it but not using PMI
– swapedoc
yesterday












@swapedoc How to prove it without PMI? Could you give me a hint? That may help.
– yuanming luo
yesterday






@swapedoc How to prove it without PMI? Could you give me a hint? That may help.
– yuanming luo
yesterday














hint: look for faulhaber formula , also try to guess coefficients of x^k in (x+1)(x+2)(x+3)....(x+n) where k can be from 0 to n
– swapedoc
yesterday




hint: look for faulhaber formula , also try to guess coefficients of x^k in (x+1)(x+2)(x+3)....(x+n) where k can be from 0 to n
– swapedoc
yesterday















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