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A340992 a(n) is the (2n)-th term of the n-fold self-convolution of the number of divisors function tau.

%I A340992 #12 Dec 13 2023 11:34:56
%S A340992 1,2,8,41,216,1172,6491,36430,206472,1179104,6774048,39107400,
%T A340992 226683903,1318427762,7690414740,44970645116,263545466456,
%U A340992 1547445069318,9101515979306,53613206171619,316243949777696,1867702439169958,11042787840419398,65357054283015120
%N A340992 a(n) is the (2n)-th term of the n-fold self-convolution of the number of divisors function tau.
%H A340992 Alois P. Heinz, <a href="/A340992/b340992.txt">Table of n, a(n) for n = 0..1281</a>
%F A340992 a(n) = [x^(2n)] (Sum_{j>=1} tau(j)*x^j)^n.
%F A340992 a(n) = A320019(2n,n).
%p A340992 b:= proc(n, k) option remember; `if`(k=0, 1,
%p A340992       `if`(k=1, numtheory[tau](n+1), (q->
%p A340992        add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A340992     end:
%p A340992 a:= n-> b(n$2):
%p A340992 seq(a(n), n=0..23);
%t A340992 T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0], If[k == 1, If[n == 0, 0, DivisorSigma[0, n]], With[{q = Quotient[k, 2]}, Sum[T[j, q]*T[n - j, k - q], {j, 0, n}]]]];
%t A340992 a[n_] := T[2n, n];
%t A340992 Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Dec 13 2023, after _Alois P. Heinz_ in A320019 *)
%Y A340992 Cf. A000005, A320019.
%K A340992 nonn
%O A340992 0,2
%A A340992 _Alois P. Heinz_, Feb 01 2021