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%I #15 Nov 20 2021 05:56:49
%S 1,3,3,6,3,9,3,8,6,9,3,18,3,9,9,9,3,18,3,18,9,9,3,24,6,9,8,18,3,27,3,
%T 9,9,9,9,36,3,9,9,24,3,27,3,18,18,9,3,27,6,18,9,18,3,24,9,24,9,9,3,54,
%U 3,9,18,9,9,27,3,18,9,27,3,48,3,9,18,18,9,27,3,27,9,9,3,54,9
%N Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = (zeta(s))^3 / (zeta(3*s))^2.
%C There is a family of multiplicative sequences based on trinomial numbers (A027907) and some fixed integer k >= 0. Let a(k,n), k >= 0, n > 0, be multiplicative with a(k,p^e) = Sum_{i=0..e} trinomial(k,i) for prime p and e >= 0, where trinomial(n,k) = 0 if 2*n < k. These sequences have the Dirichlet g.f.: Sum_{n>=1} a(k,n)/n^s = (zeta(s))^(k+1) / (zeta(3*s))^k. For several members of the family see A000012 (k=0), A073184 (k=1), and this sequence (k=2).
%H Vaclav Kotesovec, <a href="/A341343/b341343.txt">Table of n, a(n) for n = 1..10000</a>
%F Multiplicative with a(p^e) = Sum_{i=0..e} trinomial(2,i) for prime p and e >= 0, where trinomial(n,k) = 0 if 2*n < k.
%F Let b(n), n > 0, be the Dirichlet inverse of a(n); b(n) is multiplicative with b(p^(3*e)) = 1 for e >= 0, and b(p^(3*e-2)) = -3*e and b(p^(3*e-1)) = 3*e for e > 0 and prime p.
%F Sum_{k=1..n} a(k) ~ n * (log(n)^2/2 + (3*gamma - 6*zeta'(3)/zeta(3) - 1)*log(n) + 1 - 3*gamma + 3*gamma^2 + 6*(1 - 3*gamma)*zeta'(3)/zeta(3) + 27*zeta'(3)^2 / zeta(3)^2 - 9*zeta''(3)/zeta(3) - 3*sg1) / zeta(3)^2, where gamma is the Euler-Mascheroni constant A001620, zeta(3) = A002117, zeta'(3) = -A244115, zeta''(3) = A340442 and sg1 is the first Stieltjes constant (see A082633). - _Vaclav Kotesovec_, Nov 20 2021
%o (PARI) {T(n,k) = if( n<0, 0, polcoeff( (1 + x + x^2)^n, k))}; \\ A027907
%o a(n)={my(f=factor(n));prod(k=1,#f[,1],sum(i=0,f[k,2],T(2,i)))};
%o for(j=1,75,print1(a(j),", ")) \\ _Hugo Pfoertner_, Feb 13 2021
%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - X^3)^2/(1 - X)^3)[n], ", ")) \\ _Vaclav Kotesovec_, Nov 20 2021
%Y Cf. A000012, A027907, A073184.
%K nonn,easy,mult
%O 1,2
%A _Werner Schulte_, Feb 09 2021
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