A321326 -id:A321326 - cp's OEIS frontend

A321088 G.f. satisfies: A(x) = 1/(1 - x) * Product_{k>0} A(x^(2k)) / Product_{k>1} A(x^(2k-1)).

1, 1, 2, 1, 4, 2, 4, -1, 9, 3, 11, -4, 17, -2, 11, -24, 31, -3, 39, -35, 70, -14, 47, -107, 112, -27, 122, -163, 198, -90, 93, -409, 282, -108, 329, -487, 601, -160, 324, -1076, 835, -165, 907, -1298, 1478, -429, 565, -2973, 1745, -427, 1999, -3149, 3587, -528

Offset: 0

Seiichi Manyama, Nov 05 2018

Seiichi Manyama, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Convolution inverse of A321326.

Cf. A067856, A073709, A129374.

b[n_] := If[n == 1, 1, Product[{p, e} = pe; If[2 == p, e--, If[e > 1, p = 0, p = -1]]; p^e, {pe, FactorInteger[n]}]];
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b];
a = etr[b];
a /@ Range[0, 100] (* Jean-François Alcover, Oct 01 2019 *)

Euler transform of A067856.
G.f.: Product_{k>0} 1/(1 - x^k)^A067856(k).
Product_{k>0} A(x^k) = Product_{k>=0} 1/(1 - x^(2^k))^(2^k). (Cf. A073709.)

A321317 G.f. satisfies: A(x) = (1-x) * A(x^2)A(x^3)A(x^4)...A(x^n)*... .

Original entry on oeis.org

1, -1, -1, 0, -1, 2, -1, 3, -1, 1, 4, -2, -4, -1, 7, -11, 0, -6, 3, -6, 9, 2, -6, 2, 5, 12, -16, 48, 4, -1, -28, 26, -30, 30, -62, 12, 16, -55, -9, -39, 18, -47, 103, -149, 87, -182, 210, -9, 45, 75, 225, -174, 39, 273, 11, 164, -224, 77, -105, 117, -703, 715, -678

Offset: 0

Seiichi Manyama, Nov 04 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Convolution inverse of A129374.

Cf. A074206, A129373, A321326.

G.f.: Product_{k>=1} (1 - x^k)^A074206(k) where A074206(n) is the number of ordered factorizations of n.

A321325 G.f. satisfies: A(x) = (1 + x) * Product_{k>0} A(x^(2k)) / Product_{k>1} A(x^(2k-1)).

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 0, -2, 2, 1, 2, -3, 1, -3, -1, -7, 8, 4, 9, -7, 7, -7, 0, -21, 15, 2, 18, -23, 8, -25, -1, -43, 46, 17, 58, -34, 40, -41, 9, -98, 79, 10, 100, -98, 40, -123, -2, -191, 176, 43, 237, -136, 144, -192, 30, -362, 277, 12, 373, -314, 131, -457, -9, -606

Offset: 0

Seiichi Manyama, Nov 05 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A067856, A073707, A129373, A321326.

G.f.: Product_{k>0} (1 + x^k)^A067856(k).

Product_{k>0} A(x^k) = Product_{k>=0} (1 + x^(2^k))^(2^k). (Cf. A073707.)

A321327 Expansion of Product_{k>=0} (1 - x^(2^k))^(2^k).

Original entry on oeis.org

1, -1, -2, 2, -3, 3, 8, -8, -6, 6, 4, -4, 26, -26, -56, 56, -7, 7, 70, -70, -51, 51, 32, -32, 120, -120, -272, 272, -200, 200, 672, -672, -182, 182, -308, 308, 1026, -1026, -1744, 1744, -660, 660, 3064, -3064, -916, 916, -1232, 1232, 2466, -2466, -3700, 3700, -3990, 3990

Offset: 0

Seiichi Manyama, Nov 05 2018

sign

Convolution inverse of A073709.

Cf. A073707, A321326.

G.f.: A(x) satisfies A(x) = (1 - x) * A(x^2)^2, with A(0) = 1.

cp's OEIS Frontend

A321088 G.f. satisfies: A(x) = 1/(1 - x) * Product_{k>0} A(x^(2k)) / Product_{k>1} A(x^(2k-1)).

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A321317 G.f. satisfies: A(x) = (1-x) * A(x^2)A(x^3)A(x^4)...A(x^n)*... .

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A321325 G.f. satisfies: A(x) = (1 + x) * Product_{k>0} A(x^(2k)) / Product_{k>1} A(x^(2k-1)).

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Formula

A321327 Expansion of Product_{k>=0} (1 - x^(2^k))^(2^k).

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Author

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Formula

cp's OEIS Frontend

A321088 G.f. satisfies: A(x) = 1/(1 - x) * Product_{k>0} A(x^(2*k)) / Product_{k>1} A(x^(2*k-1)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A321317 G.f. satisfies: A(x) = (1-x) * A(x^2)*A(x^3)*A(x^4)*...*A(x^n)*... .

Views

Author

Keywords

Links

Crossrefs

Formula

A321325 G.f. satisfies: A(x) = (1 + x) * Product_{k>0} A(x^(2*k)) / Product_{k>1} A(x^(2*k-1)).

Views

Author

Keywords

Links

Crossrefs

Formula

A321327 Expansion of Product_{k>=0} (1 - x^(2^k))^(2^k).

Views

Author

Keywords

Crossrefs

Formula

A321088 G.f. satisfies: A(x) = 1/(1 - x) * Product_{k>0} A(x^(2k)) / Product_{k>1} A(x^(2k-1)).

A321317 G.f. satisfies: A(x) = (1-x) * A(x^2)A(x^3)A(x^4)...A(x^n)*... .

A321325 G.f. satisfies: A(x) = (1 + x) * Product_{k>0} A(x^(2k)) / Product_{k>1} A(x^(2k-1)).