cp's OEIS Frontend

A371017 The second row of generalized Knuth's old sum.

%I A371017 #27 Sep 02 2025 04:05:03
%S A371017 20,520,5880,48720,341880,2162160,12732720,71179680,382444920,
%T A371017 1991669680,10113543440,50297301600,245807780400,1183546677600,
%U A371017 5626112450400,26447537160000,123113285479800,568139770321200,2601623487262800,11830908080191200,53465154668125200,240246019677549600
%N A371017 The second row of generalized Knuth's old sum.
%H A371017 Akalu Tefera and Aklilu Zeleke, <a href="http://math.colgate.edu/~integers/x99/x99.pdf">On Proofs of Generalized Knuth's Old Sum</a>, INTEGERS, 23 (2023), #A99.
%F A371017 a(n) = 4*(8n - 3)*binomial(2n - 2,n-1)*binomial(2n+1,2n - 2)/n.
%F A371017 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).
%F A371017 b(n,m) is called a generalized Knuth's old sum.
%F A371017 G.f.: 20*x*(12*x+1)/sqrt(1-4*x)^7. - _Alois P. Heinz_, Mar 08 2024
%F A371017 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
%F A371017 a(n) ~ 2^(2*n+5) * n^(5/2) / (3*sqrt(Pi)). - _Amiram Eldar_, Sep 02 2025
%e A371017 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 ., ?, !, ∗.
%p A371017 seq(4*(8*n - 3)*binomial(2*n - 2, n-1)*binomial(2*n+1, 2*n - 2)/n, n=1..22);
%t A371017 Table[4^x*(8 x - 3)*Pochhammer[5/2, x - 1]/Pochhammer[1, x - 1], {x, 1, 25}] (* _Hugo Pfoertner_, Mar 08 2024 *)
%K A371017 nonn,changed
%O A371017 1,1
%A A371017 _Akalu Tefera_, Mar 08 2024