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 1-4 of 4 results.

A308369 G.f. A(x) satisfies: A(x) = x * Product_{k>=1} 1/(1 - A(x^k))^k.

Original entry on oeis.org

1, 1, 4, 12, 41, 133, 485, 1752, 6677, 25809, 102130, 409532, 1665128, 6837348, 28333334, 118288386, 497120101, 2101181482, 8926401690, 38093403136, 163224292328, 701951448268, 3028792691947, 13108224143298, 56887750453404, 247512117880754, 1079421026637431
Offset: 1

Views

Author

Ilya Gutkovskiy, May 22 2019

Keywords

Crossrefs

Cf. A050383, A091865, A308370, A308371, A308372.

Programs

  • Mathematica
    terms = 27; A[] = 0; Do[A[x] = x Product[1/(1 - A[x^k])^k, {k, 1, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] // Rest

Formula

G.f. A(x) satisfies: A(x) = x * exp(Sum_{k>=1} Sum_{d|k} d^2 * A(x^d)^(k/d) / k).

A308371 G.f. A(x) satisfies: A(x) = x * Product_{k>=1} 1/(1 - k*A(x^k)).

Original entry on oeis.org

1, 1, 4, 12, 42, 135, 500, 1797, 6885, 26612, 105561, 423734, 1726531, 7101261, 29486169, 123341520, 519422274, 2199966624, 9365714175, 40052639066, 171985425594, 741214499791, 3205096564624, 13901238793616, 60460193311425, 263627546862787, 1152207975128287
Offset: 1

Views

Author

Ilya Gutkovskiy, May 22 2019

Keywords

Crossrefs

Cf. A050383, A091865, A308369, A308370, A308372.

Programs

  • Mathematica
    terms = 27; A[] = 0; Do[A[x] = x Product[1/(1 - k A[x^k]), {k, 1, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] // Rest

Formula

G.f. A(x) satisfies: A(x) = x * exp(Sum_{k>=1} Sum_{d|k} d * (d * A(x^d))^(k/d) / k).

A308372 G.f. A(x) satisfies: A(x) = x * Product_{k>=1} (1 + k*A(x^k)).

Original entry on oeis.org

1, 1, 3, 8, 19, 45, 110, 259, 614, 1466, 3479, 8239, 19581, 46445, 110209, 261555, 620649, 1472597, 3494663, 8292514, 19677729, 46694303, 110804310, 262932172, 623928374, 1480555791, 3513297447, 8336903884, 19783134767, 46944538382, 111397439864, 264341463510
Offset: 1

Views

Author

Ilya Gutkovskiy, May 22 2019

Keywords

Crossrefs

Cf. A050383, A091865, A308369, A308370, A308371.

Programs

  • Mathematica
    terms = 32; A[] = 0; Do[A[x] = x Product[(1 + k A[x^k]), {k, 1, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] // Rest

Formula

G.f. A(x) satisfies: A(x) = x * exp(-Sum_{k>=1} Sum_{d|k} d * (-d * A(x^d))^(k/d) / k).

A308380 E.g.f. A(x) satisfies: A(x) = x * Product_{k>=1} (1 + A(x^k))^(1/k).

Original entry on oeis.org

1, 2, 9, 56, 455, 4224, 48391, 609104, 8814753, 140512400, 2483071481, 47387543928, 989622741367, 22107721563368, 530909919285495, 13581037512256544, 369627228319635329, 10633498287935101920, 323389433072136213289, 10342303284390333962600, 347514522157550224614711
Offset: 1

Views

Author

Ilya Gutkovskiy, May 23 2019

Keywords

Crossrefs

Cf. A091865, A308370, A308379.

Programs

  • Mathematica
    terms = 21; A[] = 0; Do[A[x] = x Product[(1 + A[x^k])^(1/k), {k, 1, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] Range[0, terms]! // Rest

Formula

E.g.f. A(x) satisfies: A(x) = x * exp(-Sum_{k>=1} Sum_{d|k} (-A(x^d))^(k/d) / k).
Showing 1-4 of 4 results.

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