A371017 The second row of generalized Knuth's old sum.
20, 520, 5880, 48720, 341880, 2162160, 12732720, 71179680, 382444920, 1991669680, 10113543440, 50297301600, 245807780400, 1183546677600, 5626112450400, 26447537160000, 123113285479800, 568139770321200, 2601623487262800, 11830908080191200, 53465154668125200, 240246019677549600
Offset: 1
Examples
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 ., ?, !, ∗.
Links
- Akalu Tefera and Aklilu Zeleke, On Proofs of Generalized Knuth's Old Sum, INTEGERS, 23 (2023), #A99.
Programs
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Maple
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
Table[4^x*(8 x - 3)*Pochhammer[5/2, x - 1]/Pochhammer[1, x - 1], {x, 1, 25}] (* Hugo Pfoertner, Mar 08 2024 *)
Formula
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.
G.f.: 20*x*(12*x+1)/sqrt(1-4*x)^7. - Alois P. Heinz, Mar 08 2024
D-finite with recurrence (-n+1)*a(n) + 2*(-4*n+21)*a(n-1) + 24*(2*n-1)*a(n-2) = 0. - R. J. Mathar, Sep 27 2024
a(n) ~ 2^(2*n+5) * n^(5/2) / (3*sqrt(Pi)). - Amiram Eldar, Sep 02 2025