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.

A307648 G.f. A(x) satisfies: 1/(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, -1, -4, -3, -2, 7, 7, 4, -6, 14, -11, -4, -47, 9, 6, 161, -93, -33, -269, 232, -83, 660, -733, 500, -779, 1527, -2291, 1876, -3892, 5598, -3056, 7791, -14088, 11289, -17113, 28083, -26211, 34645, -60715, 73180, -80951, 111926, -155269, 178561, -233709, 359679, -403884, 454659, -697310, 862133
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 19 2019

Keywords

Comments

Euler transform of A055615.

Examples

			G.f.: A(x) = 1 + x - x^2 - 4*x^3 - 3*x^4 - 2*x^5 + 7*x^6 + 7*x^7 + 4*x^8 - 6*x^9 + 14*x^10 - 11*x^11 - 4*x^12 - 47*x^13 + ...

Crossrefs

Cf. A008683, A046970, A055615, A117209, A307649.

Programs

  • Mathematica
    terms = 50; CoefficientList[Series[Product[1/(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/((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/(1 - x^k)^(mu(k)*k).
G.f.: exp(Sum_{k>=1} A046970(k)*x^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).

A307657 G.f. A(x) satisfies: 1/(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, 3, 0, 1, 10, -13, 33, -24, 26, 22, -53, 148, -181, 233, -146, 69, 157, -373, 721, -884, 1071, -1076, 1041, -444, -425, 1767, -3225, 5232, -7144, 7942, -6920, 4999, -1112, -5947, 15481, -24273, 32103, -39255, 43839, -37560, 18235, 6158, -35079, 81689, -134919, 167667, -179256
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 20 2019

Keywords

Comments

Convolution inverse of A307649.

Examples

			G.f.: A(x) = 1 - x + 3*x^2 + x^4 + 10*x^5 - 13*x^6 + 33*x^7 - 24*x^8 + 26*x^9 + 22*x^10 - 53*x^11 + 148*x^12 - 181*x^13 + ...

Crossrefs

Cf. A008683, A055615, A117211, A307649.

Programs

  • Mathematica
    terms = 47; CoefficientList[Series[Product[1/(1 + x^k)^(MoebiusMu[k] k), {k, 1, terms}], {x, 0, terms}], x]
terms = 47; A[] = 1; Do[A[x] = 1/((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/(1 + x^k)^(mu(k)*k).
Showing 1-3 of 3 results.

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