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 1-5 of 5 results.

A307411 G.f. A(x) satisfies: A(x) = 1 + x*A(x)*(1 + 2*x*A(x))/(1 - x*A(x) - x^2*A(x)^2).

Original entry on oeis.org

1, 1, 4, 14, 60, 267, 1254, 6071, 30156, 152714, 785682, 4094752, 21573258, 114709363, 614777462, 3317589966, 18011350796, 98307220409, 539121535194, 2969177051678, 16415395615190, 91070109305056, 506843759000184, 2828968117483929, 15831944500607010, 88818114923080102
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 07 2019

Keywords

Crossrefs

Cf. A000204, A049124, A307412, A307413.

Programs

  • Mathematica
    terms = 25; A[] = 0; Do[A[x] = 1 + x A[x] (1 + 2 x A[x])/(1 - 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[LucasL[k] 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 - x - x^2)/(1 + x^2), {x, 0, terms}], x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} Lucas(k)*x^k*A(x)^k, where Lucas = A000204.
G.f.: A(x) = (1/x)*Series_Reversion(x*(1 - x - x^2)/(1 + x^2)).

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

A326564 O.g.f. A(x) satisfies: 0 = Sum_{n>=1} (b(n) - A(x))^n * (2*x)^n / n, where b(n) = 1 if n is odd or b(n) = 2 if n is even.

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, 61838364112438732, -306647601790749384, 1523380558254732312, -7580755340625743760, 37783723921640161923
Offset: 0

Views

Author

Paul D. Hanna, Aug 28 2019

Keywords

Comments

a(n) is odd iff n = 2^k - 1 for k >= 0.
Signed version of A307413.

Examples

			O.g.f.: A(x) = 1 + x - 2*x^2 + 7*x^3 - 26*x^4 + 102*x^5 - 420*x^6 + 1787*x^7 - 7794*x^8 + 34666*x^9 - 156636*x^10 + 716982*x^11 - 3317700*x^12 + 15494156*x^13 - 72935624*x^14 + 345701843*x^15 - 1648489762*x^16 + ...
such that
0 = (1 - A(x))*(2*x) + (2 - A(x))^2*(2*x)^2/2 + (1 - A(x))^3*(2*x)^3/3 + (2 - A(x))^4*(2*x)^4/4 + (1 - A(x))^5*(2*x)^5/5 + (2 - A(x))^6*(2*x)^6/6 + (1 - A(x))^7*(2*x)^7/7 + (2 - A(x))^8*(2*x)^8/8 + (1 - A(x))^9*(2*x)^9/9 + ...
SPECIAL ARGUMENTS.
A( (3 - sqrt(17))/6 ) = 1/2.
A( (15 - sqrt(513))/40 ) = 1/3.
ODD TERMS.
The odd numbers occur at positions 2^n-1 and begin
[1, 1, 7, 1787, 345701843, 37783723921640161923, 1297226675901009799785880946943488094880739, 4359630365907394639251834255689265800511483817161978056491648421720696612963282942355107, ...].

Links

Crossrefs

Cf. A316363, A307413.

Programs

  • PARI
    /* By definition */
{a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0); A[#A] = polcoeff(sum(m=1, #A, ( ((m+1)%2) + 1 - Ser(A) )^m * (2*x)^m/m), #A)/2); A[n+1]}
for(n=0, 32, print1(a(n), ", "))
  • PARI
    /* From: A(x) = 2 - (1/x) * Series_Reversion( x + x^2/(1 - 2*x^2) ) */
{a(n) = my(A = 2 - (1/x)*serreverse(x + x^2/(1 - 2*x^2 +x*O(x^n)))); polcoeff(A,n)}
for(n=0, 32, print1(a(n), ", "))

Formula

O.g.f. A = A(x) satisfies:
(1) 0 = Sum_{n>=1} (3 + (-1)^n - 2*A(x))^n * x^n / n.
(2) 0 = arctanh(2*x - 2*x*A) - log(1 - 4*x^2*(2 - A)^2)/2.
(3) 1 - 4*x^2*(2 - A)^2 = (1 + 2*x - 2*x*A) / (1 - 2*x + 2*x*A).
(4) A(x) = 1 + (A - 2)^2*x + 2*(A - 1)*(A - 2)^2*x^2.
(5) 0 = 2*(A - 1)*(A - 2)^2*x^2 + (A - 2)^2*x - (A - 1).
(6) x = ( sqrt( (A-2)^4 + 8*(A-1)^2*(A-2)^2 ) - (A-2)^2 ) / (4*(A-1)*(A-2)^2).
(7) A(x) = 2 - (1/x) * Series_Reversion( x + x^2/(1 - 2*x^2) ).

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 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.