cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A307649 G.f. A(x) satisfies: (1 + x) = A(x)*A(x^2)^2*A(x^3)^3*A(x^4)^4* ... *A(x^k)^k* ...

Original entry on oeis.org

1, 1, -2, -5, 0, 4, 9, 2, -10, -21, 29, 15, -18, -80, 50, 59, 207, -228, -244, -315, 868, 103, 360, -1907, 752, -151, 3802, -5032, 965, -5279, 13742, -6049, 9107, -33835, 25398, -15098, 63365, -79614, 51752, -117194, 196980, -156321, 209085, -435223, 463497, -441950, 871202, -1146187, 1023944, -1704179
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 19 2019

Keywords

Comments

Weigh transform of A055615.

Examples

			G.f.: A(x) = 1 + x - 2*x^2 - 5*x^3 + 4*x^5 + 9*x^6 + 2*x^7 - 10*x^8 - 21*x^9 + 29*x^10 + 15*x^11 - 18*x^12 - 80*x^13 + ...

Crossrefs

Cf. A008683, A055615, A117210, A307648.

Programs

  • Mathematica
    terms = 49; CoefficientList[Series[Product[(1 + x^k)^(MoebiusMu[k] k), {k, 1, terms}], {x, 0, terms}], x]
terms = 49; A[] = 1; Do[A[x] = (1 + x)/Product[A[x^k]^k, {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

Formula

G.f.: Product_{k>=1} (1 + x^k)^(mu(k)*k).

A307659 G.f. A(x) satisfies: (1 + x)/(1 - x) = A(x)*A(x^2)^2*A(x^3)^3*A(x^4)^4* ... *A(x^k)^k* ...

Original entry on oeis.org

1, 2, -2, -12, -10, 12, 44, 40, -26, -132, -60, 104, 156, -152, -136, 80, 1030, 204, -1548, -3736, 1780, 5096, 7496, -8208, -9956, -11416, 30824, 2784, -296, -72768, 62032, 22752, 157126, -290356, -41420, -244296, 896996, -285896, 436040, -2139120, 1661940, -1028632, 4535608, -6487744, 4125224
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 20 2019

Keywords

Comments

Convolution of A307648 and A307649.

Examples

			G.f.: A(x) = 1 + 2*x - 2*x^2 - 12*x^3 - 10*x^4 + 12*x^5 + 44*x^6 + 40*x^7 - 26*x^8 - 132*x^9 - 60*x^10 + ...

Crossrefs

Cf. A008683, A055615, A307648, A307649, A307658.

Programs

  • Mathematica
    terms = 44; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(MoebiusMu[k] k), {k, 1, terms}], {x, 0, terms}], x]
terms = 44; A[] = 1; Do[A[x] = (1 + x)/((1 - x) Product[A[x^k]^k, {k, 2, terms}]) + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

Formula

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^(mu(k)*k).

A307656 G.f. A(x) satisfies: (1 - x) = A(x)*A(x^2)^2*A(x^3)^3*A(x^4)^4* ... *A(x^k)^k* ...

Original entry on oeis.org

1, -1, 2, 1, 0, 8, -7, 22, -6, 13, 29, -11, 82, -36, 114, 13, 103, 88, 88, 275, -20, 549, -200, 1007, -144, 811, 730, 188, 2093, -777, 3538, -643, 4083, -537, 4562, 2478, 1973, 8062, -3508, 17362, -8164, 20281, -2227, 17483, 8605, 2946, 30190, -6085, 53176, -28913, 78516
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 20 2019

Keywords

Comments

Convolution inverse of A307648.

Examples

			G.f.: A(x) = 1 - x + 2*x^2 + x^3 + 8*x^5 - 7*x^6 + 22 x^7 - 6*x^8 + 13*x^9 + 29*x^10 - 11*x^11 + 82*x^12 - 36*x^13 + ...

Crossrefs

Cf. A008683, A046970, A055615, A117208, A307648.

Programs

  • Mathematica
    terms = 50; CoefficientList[Series[Product[(1 - x^k)^(MoebiusMu[k] k), {k, 1, terms}], {x, 0, terms}], x]
terms = 50; CoefficientList[Series[Exp[-Sum[Sum[MoebiusMu[d] d^2, {d, Divisors[k]}] x^k/k, {k, 1, terms}]], {x, 0, terms}], x]
terms = 50; A[] = 1; Do[A[x] = (1 - x)/Product[A[x^k]^k, {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

Formula

G.f.: Product_{k>=1} (1 - x^k)^(mu(k)*k).
G.f.: exp(-Sum_{k>=1} A046970(k)*x^k/k).
Showing 1-3 of 3 results.

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