A338045 -id:A338045 - cp's OEIS frontend

A338046 G.f.: Sum_{k>=0} x^(2^k) / (1 - x^(2^k))^4.

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

Ilya Gutkovskiy, Oct 08 2020

nonn

Cf. A000292, A000335, A000447, A001511, A002492, A129527, A209229, A328407, A338045.

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}]

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.

A373186 Expansion of Sum_{k>=0} x^(3^k) / (1 - x^(3^k))^3.

Original entry on oeis.org

1, 3, 7, 10, 15, 24, 28, 36, 52, 55, 66, 88, 91, 105, 135, 136, 153, 195, 190, 210, 259, 253, 276, 336, 325, 351, 430, 406, 435, 520, 496, 528, 627, 595, 630, 754, 703, 741, 871, 820, 861, 1008, 946, 990, 1170, 1081, 1128, 1312, 1225, 1275, 1479, 1378, 1431, 1680, 1540

Offset: 1

Seiichi Manyama, May 27 2024

nonn

Cf. A129527, A373187.
Cf. A327625, A338045.
Cf. A000217.

G.f. A(x) satisfies A(x) = x/(1 - x)^3 + A(x^3).

a(3*n+1) = A000217(3*n+1), a(3*n+2) = A000217(3*n+2) and a(3*n+3) = A000217(3*n+3) + a(n+1) for n >= 0.

cp's OEIS Frontend

A338046 G.f.: Sum_{k>=0} x^(2^k) / (1 - x^(2^k))^4.

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