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A371017
The second row of generalized Knuth's old sum.
0
20, 520, 5880, 48720, 341880, 2162160, 12732720, 71179680, 382444920, 1991669680, 10113543440, 50297301600, 245807780400, 1183546677600, 5626112450400, 26447537160000, 123113285479800, 568139770321200, 2601623487262800, 11830908080191200, 53465154668125200, 240246019677549600
OFFSET
1,1
LINKS
A. Tefera and A. Zeleke, On Proofs of Generalized Knuth's Old Sum, INTEGERS, 23 (2023), #A99.
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
EXAMPLE
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 ., ?, !, ∗.
MAPLE
seq(4*(8*n - 3)*binomial(2*n - 2, n-1)*binomial(2*n+1, 2*n - 2)/n, n=1..22);
MATHEMATICA
Table[4^x*(8 x - 3)*Pochhammer[5/2, x - 1]/Pochhammer[1, x - 1], {x, 1, 25}] (* Hugo Pfoertner, Mar 08 2024 *)
CROSSREFS
Sequence in context: A130186 A130033 A250016 * A337352 A180882 A202920
KEYWORD
nonn
AUTHOR
Akalu Tefera, Mar 08 2024
STATUS
approved