A308418 -id:A308418 - cp's OEIS frontend

A308422 a(n) = n^2 if n odd, 3*n^2/4 if n even.

0, 1, 3, 9, 12, 25, 27, 49, 48, 81, 75, 121, 108, 169, 147, 225, 192, 289, 243, 361, 300, 441, 363, 529, 432, 625, 507, 729, 588, 841, 675, 961, 768, 1089, 867, 1225, 972, 1369, 1083, 1521, 1200, 1681, 1323, 1849, 1452, 2025, 1587, 2209, 1728, 2401, 1875, 2601, 2028, 2809, 2187, 3025

Offset: 0

Ilya Gutkovskiy, May 26 2019

Moebius transform of A076577.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A000290, A007434, A016754, A026741, A033428, A076577, A308418, A309337.

a[n_] := If[OddQ[n], n^2, 3 n^2/4]; Table[a[n], {n, 0, 55}]
nmax = 55; CoefficientList[Series[x (1 + 3 x + 6 x^2 + 3 x^3 + x^4)/(1 - x^2)^3, {x, 0, nmax}], x]
LinearRecurrence[{0, 3, 0, -3, 0, 1}, {0, 1, 3, 9, 12, 25}, 56]
Table[(7 - (-1)^n) n^2/8, {n, 0, 55}]

G.f.: x*(1 + 3*x + 6*x^2 + 3*x^3 + x^4)/(1 - x^2)^3.
G.f.: Sum_{k>=1} J_2(k)*x^k/(1 - x^(2*k)), where J_2() is the Jordan function (A007434).
E.g.f.: x*((4 + 3*x)*cosh(x) + (3 + 4*x)*sinh(x))/4.
Dirichlet g.f.: zeta(s-2)*(1 - 1/2^s).
a(n) = (7 - (-1)^n)*n^2/8.
a(n) = Sum_{d|n, n/d odd} J_2(d).
a(2*k+1) = A016754(k), a(2*k) = A033428(k).
Sum_{n>=1} 1/a(n) = 13*Pi^2/72 = 1.7820119057522453061...
Sum_{n>=1} (-1)^(n+1)/a(n) = 5*Pi^2/72 = 0.68538919452009434853...
Multiplicative with a(2^e) = 3*2^(2*e-2), and a(p^e) = p^(2*e) for odd primes p. - Amiram Eldar, Oct 26 2020
For n >= 1, n*a(n) = A309337(n) = Sum_{d divides n} (-1)^(d+1) * J(3, n/d), where the Jordan totient function J_3(n) = A059376. - Peter Bala, Jan 21 2024

A308417 Expansion of e.g.f. exp(x*(1 + x + x^2)/(1 - x^2)^2).

Original entry on oeis.org

1, 1, 3, 25, 145, 1461, 14011, 169933, 2231265, 32572585, 528302611, 9146070561, 174016032433, 3498446485405, 75954922790475, 1737982233878101, 42327522277348801, 1084073452000879953, 29253450397798616995, 827617575903336189865, 24503022168956714812881

Offset: 0

Ilya Gutkovskiy, May 25 2019

nonn

Cf. A007434, A026741, A082579, A088009, A301876, A308418.

nmax = 20; CoefficientList[Series[Exp[x (1 + x + x^2)/(1 - x^2)^2], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Product[(1 + x^k)^(DirichletConvolve[j^2, MoebiusMu[j], j, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Numerator[k/2] k! Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 20}]

my(x ='x + O('x^30)); Vec(serlaplace(exp(x*(1 + x + x^2)/(1 - x^2)^2))) \\ Michel Marcus, May 26 2019

E.g.f.: exp(Sum_{k>=1} A026741(k)*x^k).
E.g.f.: Product_{k>=1} (1 + x^k)^(J_2(k)/k), where J_2() is the Jordan function (A007434).
a(0) = 1; a(n) = Sum_{k=1..n} A026741(k)*k!*binomial(n-1,k-1)*a(n-k).
a(n) ~ 2^(-1/6) * 3^(-1/3) * n^(n - 1/6) * exp((3/2)^(4/3) * n^(2/3) - n). - Vaclav Kotesovec, May 29 2019

cp's OEIS Frontend

A308422 a(n) = n^2 if n odd, 3*n^2/4 if n even.

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