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A193446 a(n) = n! * Sum_{k=1..n-1} H(k)*H(n-k) for n>=2, where H(n) is the n-th harmonic number.

%I A193446 #21 Feb 16 2025 08:33:15
%S A193446 2,18,142,1160,10208,97720,1018008,11507472,140581872,1848236544,
%T A193446 26041957632,391738883328,6268657962240,106361504663040,
%U A193446 1907768564282880,36075270482058240,717379367864693760,14967427450111488000,326956068509511168000,7463341244426167296000
%N A193446 a(n) = n! * Sum_{k=1..n-1} H(k)*H(n-k) for n>=2, where H(n) is the n-th harmonic number.
%C A193446 A self-convolution of the harmonic numbers.
%H A193446 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>.
%F A193446 E.g.f.: log(1 - x)^2 / (1 - x)^2.
%F A193446 a(n) = n! * Sum_{k=1..n-1} (Sum_{i=1..k} 1/i)*(Sum_{j=1..n-k} 1/j) for n>=2.
%e A193446 E.g.f.: log(1-x)^2/(1-x)^2 = 2*x^2/2! + 18*x^3/3! + 142*x^4/4! + 1160*x^5/5! + ...
%e A193446 Illustration of initial terms:
%e A193446 a(2) = 2!*(1*1) = 2;
%e A193446 a(3) = 3!*(1*(1+1/2) + (1+1/2)*1) = 18;
%e A193446 a(4) = 4!*(1*(1+1/2+1/3) + (1+1/2)*(1+1/2) + (1+1/2+1/3)*1) = 142;
%e A193446 a(5) = 5!*(1*(1+1/2+1/3+1/4) + (1+1/2)*(1+1/2+1/3) + (1+1/2+1/3)*(1+1/2) + (1+1/2+1/3+1/4)*1) = 1160; ...
%t A193446 a[n_] := n! * Sum[HarmonicNumber[k] * HarmonicNumber[n-k], {k, 1, n-1}]; Array[a, 20, 2] (* _Amiram Eldar_, Aug 18 2023 *)
%o A193446 (PARI) {a(n)=if(n<2,0,n!*sum(k=1,n,sum(i=1,k,1/i)*sum(j=1,n-k,1/j)))}
%o A193446 (PARI) {a(n)=if(n<0,0,n!*polcoeff(log(1-x+x*O(x^n))^2/(1-x+x*O(x^n))^2,n))}
%Y A193446 Cf. A001008, A002805, A001706, A193445.
%K A193446 nonn
%O A193446 2,1
%A A193446 _Paul D. Hanna_, Jul 26 2011
%E A193446 a(21) from _Amiram Eldar_, Aug 18 2023