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A338046
G.f.: Sum_{k>=0} x^(2^k) / (1 - x^(2^k))^4.
1
1, 5, 10, 25, 35, 66, 84, 145, 165, 255, 286, 430, 455, 644, 680, 961, 969, 1305, 1330, 1795, 1771, 2310, 2300, 3030, 2925, 3731, 3654, 4704, 4495, 5640, 5456, 6945, 6545, 8109, 7770, 9741, 9139, 11210, 10660, 13275, 12341, 15015, 14190, 17490, 16215, 19596, 18424, 22630
OFFSET
1,2
FORMULA
G.f. A(x) satisfies: A(x) = A(x^2) + x / (1 - x)^4.
a(2*n) = a(n) + A002492(n), a(2*n+1) = A000447(n+1).
a(n) = (1/6) * Sum_{d|n} A209229(n/d) * d * (d + 1) * (d + 2).
Product_{n>=1} (1 + x^n)^a(n) = g.f. for A000335.
MATHEMATICA
nmax = 48; CoefficientList[Series[Sum[x^(2^k) /(1 - x^(2^k))^4, {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x] // Rest
a[n_] := If[EvenQ[n], a[n/2] + n (n + 1) (n + 2)/6, n (n + 1) (n + 2)/6]; Table[a[n], {n, 1, 48}]
Table[(1/6) DivisorSum[n, Boole[IntegerQ[Log[2, n/#]]] # (# + 1) (# + 2) &], {n, 1, 48}]
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Oct 08 2020
STATUS
approved