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A371017
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The second row of generalized Knuth's old sum.
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0
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20, 520, 5880, 48720, 341880, 2162160, 12732720, 71179680, 382444920, 1991669680, 10113543440, 50297301600, 245807780400, 1183546677600, 5626112450400, 26447537160000, 123113285479800, 568139770321200, 2601623487262800, 11830908080191200, 53465154668125200, 240246019677549600
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OFFSET
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1,1
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LINKS
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FORMULA
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a(n) = 4*(8n - 3)*binomial(2n - 2,n-1)*binomial(2n+1,2n - 2)/n.
a(n) = b(n+1,1), where b(n,m) = Sum_{i=0..m} binomial(2n+2m+1,2n) * binomial(2m+1,2i+1) * binomial(2n+2m-2i,n+m-i)*2^(2i+1) = Sum_{k=0..2n} (-1)^k * binomial(4m+2,2m+1) * binomial(2n+2m+1,k+2m+1) * binomial(2k,k)*2^(2n-k).
b(n,m) is called a generalized Knuth's old sum.
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EXAMPLE
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For fixed n, consider a set S of words in the alphabet {a, b, c, d, C, D, ?, ., !, ∗} such that w is in S if and only if the following conditions all hold: (i) the number of letters in w is 2n; (ii) the number of a's in w is equal to the number of b's in w; (iii) the number of special characters in w is 3; (iv) the number of .'s in w is equal to the number of ?'s in w. Then a(n) is the number of words w in S such that w has only lowercase letters a, b, and the special characters ., ?, !, ∗.
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MAPLE
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seq(4*(8*n - 3)*binomial(2*n - 2, n-1)*binomial(2*n+1, 2*n - 2)/n, n=1..22);
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MATHEMATICA
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Table[4^x*(8 x - 3)*Pochhammer[5/2, x - 1]/Pochhammer[1, x - 1], {x, 1, 25}] (* Hugo Pfoertner, Mar 08 2024 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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