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.

A307413 G.f. A(x) satisfies: A(x) = 1 + x*A(x)/(1 - x*A(x) - 2*x^2*A(x)^2).

Original entry on oeis.org

1, 1, 2, 7, 26, 102, 420, 1787, 7794, 34666, 156636, 716982, 3317700, 15494156, 72935624, 345701843, 1648489762, 7902956738, 38067806892, 184152092450, 894259126540, 4357738501844, 21302682030328, 104439435098718, 513390992000340, 2529846489669412, 12494572784556440
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 07 2019

Keywords

Comments

Unsigned version of A326564. - Paul D. Hanna, Aug 28 2019

Crossrefs

Cf. A001045, A049124, A307411, A307412.
Cf. A326564.

Programs

  • Mathematica
    terms = 26; A[] = 0; Do[A[x] = 1 + x A[x]/(1 - x A[x] - 2 x^2 A[x]^2) + O[x]^(terms + 1) // Normal, {terms + 1}]; CoefficientList[A[x], x]
terms = 27; A[] = 0; Do[A[x] = 1 + Sum[(1/3) (2^k - (-1)^k) x^k A[x]^k, {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
terms = 27; CoefficientList[1/x InverseSeries[Series[x (1 + x) (1 - 2 x)/(1 - 2 x^2), {x, 0, terms}], x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} Jacobsthal(k)*x^k*A(x)^k, where Jacobsthal = A001045.
G.f.: A(x) = (1/x)*Series_Reversion(x*(1 + x)*(1 - 2*x)/(1 - 2*x^2)).
From Paul D. Hanna, Aug 29 2019: (Start)
G.f. A(x) satisfies: 0 = Sum_{n>=1} (1-(-1)^n - 2*A(x))^n * x^n / n.
G.f. A(x) satisfies: log(1 - 4*x^2*A(x)^2)/2 = arctanh(2*x - 2*x*A(x)). (End)

A307412 G.f. A(x) satisfies: A(x) = 1 + x*A(x)/(1 - 2*x*A(x) - x^2*A(x)^2).

Original entry on oeis.org

1, 1, 3, 12, 53, 250, 1234, 6295, 32925, 175616, 951596, 5223658, 28987546, 162349759, 916502869, 5209630108, 29792226533, 171284524184, 989460348216, 5740230703588, 33429379234924, 195361236443008, 1145312096390408, 6733896357727242, 39697441350016170, 234596104853541967
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 07 2019

Keywords

Crossrefs

Cf. A000129, A049124, A307411, A307413.

Programs

  • Mathematica
    terms = 25; A[] = 0; Do[A[x] = 1 + x A[x]/(1 - 2 x A[x] - x^2 A[x]^2) + O[x]^(terms + 1) // Normal, {terms + 1}]; CoefficientList[A[x], x]
terms = 26; A[] = 0; Do[A[x] = 1 + Sum[Fibonacci[k, 2] x^k A[x]^k, {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
terms = 26; CoefficientList[1/x InverseSeries[Series[x (1 - 2 x - x^2)/(1 - x - x^2), {x, 0, terms}], x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} Pell(k)*x^k*A(x)^k, where Pell = A000129.
G.f.: A(x) = (1/x)*Series_Reversion(x*(1 - 2*x - x^2)/(1 - x - x^2)).
a(n) ~ sqrt((1 + 2^(1/3))*(4 + 7*2^(1/3))) * (2 + 3/2^(2/3) + 3/2^(1/3))^n / (3 * sqrt(Pi) * (2*n)^(3/2)). - Vaclav Kotesovec, Nov 05 2021

A307528 G.f. A(x) satisfies: A(x) = 1 + x^2*A(x)^2/(1 - x*A(x) - x^2*A(x)^2 - x^3*A(x)^3).

Original entry on oeis.org

1, 0, 1, 1, 4, 9, 27, 76, 226, 680, 2078, 6441, 20153, 63684, 202732, 649930, 2095854, 6794684, 22131765, 72393439, 237703654, 783198068, 2588645047, 8580674778, 28517805357, 95009277576, 317242351135, 1061500510809, 3558683892258, 11952025977378, 40209157279701
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 12 2019

Keywords

Examples

			G.f.: A(x) = 1 + x^2 + x^3 + 4*x^4 + 9*x^5 + 27*x^6 + 76*x^7 + 226*x^8 + 680*x^9 + 2078*x^10 + ...

Crossrefs

Cf. A000073, A049124, A049140, A307411, A307412, A307413, A307529.

Programs

  • Mathematica
    terms = 31; CoefficientList[1/x InverseSeries[Series[x (1 - x - x^2 - x^3)/(1 - x - x^3), {x, 0, terms}], x], x]
terms = 30; A[] = 0; Do[A[x] = 1 + x^2 A[x]^2/(1 - x A[x] - x^2 A[x]^2 - x^3 A[x]^3) + O[x]^(terms + 1) // Normal, {terms + 1}]; CoefficientList[A[x], x]
terms = 31; t[n_] := t[n] = SeriesCoefficient[x^2/(1 - x - x^2 - x^3), {x, 0, n}]; A[] = 0; Do[A[x] = 1 + Sum[t[k] x^k A[x]^k, {k, 2, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=2} A000073(k)*x^k*A(x)^k.
G.f.: A(x) = (1/x)*Series_Reversion(x*(1 - x - x^2 - x^3)/(1 - x - x^3)).

A307529 G.f. A(x) satisfies: A(x) = (1 - x^2*A(x)^2)/(1 - x^2*A(x)^2 - x^3*A(x)^3).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 4, 1, 10, 23, 18, 92, 168, 241, 856, 1480, 2904, 8266, 14854, 33496, 83578, 161047, 380488, 884326, 1819714, 4321045, 9730466, 21019404, 49456092, 110408981, 246005440, 572574553, 1281705752, 2906696339, 6711882928, 15128432758, 34625418170
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 12 2019

Keywords

Examples

			G.f.: A(x) = 1 + x^3 + x^5 + 4*x^6 + x^7 + 10*x^8 + 23*x^9 + 18*x^10 + 92*x^11 + 168*x^12 + ...

Crossrefs

Cf. A000931, A049124, A049140, A067955, A307411, A307412, A307413, A307528.

Programs

  • Mathematica
    terms = 37; CoefficientList[1/x InverseSeries[Series[x (1 - x^2 - x^3)/(1 - x^2), {x, 0, terms}], x], x]
terms = 36; A[] = 0; Do[A[x] = (1 - x^2 A[x]^2)/(1 - x^2 A[x]^2 - x^3 A[x]^3) + O[x]^(terms + 1) // Normal, {terms + 1}]; CoefficientList[A[x], x]
terms = 37; p[n_] := p[n] = SeriesCoefficient[(1 - x^2)/(1 - x^2 - x^3), {x, 0, n}]; A[] = 1; Do[A[x] = Sum[p[k] x^k A[x]^k, {k, 0, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = Sum_{k>=0} A000931(k)*x^k*A(x)^k.
G.f.: A(x) = (1/x)*Series_Reversion(x*(1 - x^2 - x^3)/(1 - x^2)).
Showing 1-4 of 4 results.

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