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

1,1

Links

Table of n, a(n) for n=1..22.

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

Adjacent sequences: A371014 A371015 A371016 * A371018 A371019 A371020

Keyword

nonn

Author

Akalu Tefera, Mar 08 2024

Status

approved