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a(n) is the (2n)-th term of the n-fold self-convolution of the number of divisors function tau.
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%I #12 Dec 13 2023 11:34:56

%S 1,2,8,41,216,1172,6491,36430,206472,1179104,6774048,39107400,

%T 226683903,1318427762,7690414740,44970645116,263545466456,

%U 1547445069318,9101515979306,53613206171619,316243949777696,1867702439169958,11042787840419398,65357054283015120

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

%H Alois P. Heinz, <a href="/A340992/b340992.txt">Table of n, a(n) for n = 0..1281</a>

%F a(n) = [x^(2n)] (Sum_{j>=1} tau(j)*x^j)^n.

%F a(n) = A320019(2n,n).

%p b:= proc(n, k) option remember; `if`(k=0, 1,

%p `if`(k=1, numtheory[tau](n+1), (q->

%p add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))

%p end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..23);

%t 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 a[n_] := T[2n, n];

%t Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Dec 13 2023, after _Alois P. Heinz_ in A320019 *)

%Y Cf. A000005, A320019.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Feb 01 2021