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A341343
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Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = (zeta(s))^3 / (zeta(3*s))^2.
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1
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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, 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, 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
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OFFSET
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1,2
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COMMENTS
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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).
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LINKS
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FORMULA
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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.
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.
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
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PROG
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(PARI) {T(n, k) = if( n<0, 0, polcoeff( (1 + x + x^2)^n, k))}; \\ A027907
a(n)={my(f=factor(n)); prod(k=1, #f[, 1], sum(i=0, f[k, 2], T(2, i)))};
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - X^3)^2/(1 - X)^3)[n], ", ")) \\ Vaclav Kotesovec, Nov 20 2021
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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