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F(n+1) = Σ[i=0...n] Combination(n,i) * F(i) * (i+1) * F(n-i)
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Σ[k=0...n] nCk * (k+1)^k * (n-k+1)^(n-k-1) = (n+2)^n
Σ[k=0...n] nCk * (k+1)^(k-1) * (n-k+1)^(n-k-1) = 2*(n+2)^(n-1)
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https://en.m.wikipedia.org/wiki/Double_counting_(proof_technique)
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ããã§ãæé·æ°åã®ç·æ°ãã解決ããããšã«ãªããŸãã
ãããã¯ãèãæ¹ãæµçšããã°ããŸã¡éã®æšãã®åé¡ã«åž°çããããŸã§ããªããã£ã¡ã解ããããã
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ãããŠæ®ã£ã No.1051 ã®æçåŒã®è¬ã
çµã¿åãããçšããŠèšŒæã§ããããšã«ãªããŸãããåŒå€åœ¢ã§ç€ºããã®ãã©ããã
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> ãããŠæ®ã£ã No.1051 ã®æçåŒã®è¬ã
> çµã¿åãããçšããŠèšŒæã§ããããšã«ãªããŸãããåŒå€åœ¢ã§ç€ºããã®ãã©ããã
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nCk ã C[n,k] ãšæžãããšã«ããŸãã
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C[n,i] * C[n-i,j] = C[n,j] * C[n-j,i]
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f(k)ãkã®n-1次以äžã®å€é
åŒãšããŠã
Σ[k=0...n] C[n,k] * (-1)^k * f(k) = 0
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Σ[h=0...n] C[n,h] * (h+1)^h * (n-h+1)^(n-h-1)
= Σ[h=0...n] C[n,n-h] * (n-h+1)^(n-h-1) * (h+1)^h
= Σ[k=0...n] C[n,k] * (k+1)^(k-1) * (n-k+1)^(n-k)
= Σ[k=0...n] C[n,k] * (k+1)^(k-1) * ((n+2)-(k+1))^(n-k)
= Σ[k=0...n] C[n,k] * (k+1)^(k-1) * { Σ[m=0...n-k] C[n-k,m] * (n+2)^m * (-1)^(n-k-m) * (k+1)^(n-k-m) }
= Σ[k=0...n] Σ[m=0...n-k] C[n,k] * C[n-k,m] * (n+2)^m * (-1)^(n-k-m) * (k+1)^(n-m-1)
= Σ[m=0...n] Σ[k=0...n-m] C[n,k] * C[n-k,m] * (n+2)^m * (-1)^(n-k-m) * (k+1)^(n-m-1)
= (n+2)^n + Σ[m=0...n-1] Σ[k=0...n-m] C[n,k] * C[n-k,m] * (n+2)^m * (-1)^(n-k-m) * (k+1)^(n-m-1)
= (n+2)^n + Σ[m=0...n-1] Σ[k=0...n-m] C[n,m] * C[n-m,k] * (n+2)^m * (-1)^(n-m) * (-1)^k * (k+1)^(n-m-1)
= (n+2)^n + Σ[m=0...n-1] C[n,m] * (n+2)^m * (-1)^(n-m) * { Σ[k=0...n-m] C[n-m,k] * (-1)^k * (k+1)^(n-m-1) }
= (n+2)^n + Σ[m=0...n-1] C[n,m] * (n+2)^m * (-1)^(n-m) * 0
= (n+2)^n
ããããªãã»ã©ããããªæ¹æ³ã§ k+1 ã®ææ°ãã k ãæ¶ãããšã¯ã
ããã§
> Σ[k=0...n] nCk * (k+1)^k * (n-k+1)^(n-k-1) = (n+2)^n
>
> Σ[k=0...n] nCk * (k+1)^(k-1) * (n-k+1)^(n-k-1) = 2*(n+2)^(n-1)
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(a+b)^n / a = Σ[k=0...n] C[n,k] * (a+k)^(k-1) * (b-k)^(n-k)
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