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.

A302701 O.g.f. A(x) satisfies: A(x) = 1 + Integral (x/A(x))' / (x/A(x)^4)' dx.

Original entry on oeis.org

1, 1, 3, 16, 118, 1050, 10509, 113892, 1307043, 15661024, 194075098, 2470848492, 32161635070, 426440290744, 5743575712131, 78405535427220, 1082876597440146, 15109514661352482, 212736976140479073, 3019422091269739704, 43164665664066028062, 621078277521084894978, 8989001884449529431990, 130795752983608734209604, 1912460927749734257739153, 28088780052768915388505436, 414247711043291214286003410
Offset: 0

Views

Author

Paul D. Hanna, Apr 19 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 118*x^4 + 1050*x^5 + 10509*x^6 + 113892*x^7 + 1307043*x^8 + 15661024*x^9 + 194075098*x^10 + ...
RELATED SERIES.
(x/A(x))' / (x/A(x)^4)' = 1 + 6*x + 48*x^2 + 472*x^3 + 5250*x^4 + 63054*x^5 + 797244*x^6 + 10456344*x^7 + 140949216*x^8 + 1940750980*x^9 + ...
which equals A'(x).
The logarithmic derivative of the g.f. begins:
A'(x)/A(x) = 1 + 5*x + 40*x^2 + 401*x^3 + 4531*x^4 + 55040*x^5 + 701716*x^6 + 9261257*x^7 + 125449600*x^8 + 1734071855*x^9 + 24362189248*x^10 + ...
which equals (1 + x*A(x)^2 - sqrt(1 - 14*x*A(x)^2 + x^2*A(x)^4))/(8*x).

Links

Crossrefs

Cf. A302704, A302705, A303064, A338163, A338187, A338188, A338193, A338194.

Programs

  • Mathematica
    nmax = 30; A = 1; Do[A = 1 + Integrate[D[x/A, x]/D[x/A^4, x], x] + O[x]^nmax, nmax]; CoefficientList[A, x] (* Vaclav Kotesovec, Oct 15 2020 *)
  • PARI
    {a(n) = my(A=1); for(i=1, n, A = 1 + intformal( (x/A)'/(x/A^4 +x*O(x^n))' ); ); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

Formula

O.g.f. A(x) satisfies:
(1) A(x) = 1 + Integral (x/A(x))' / (x/A(x)^4)' dx.
(2) A(x) = 1 + Integral A(x)^3 * (A(x) - x*A'(x)) / (A(x) - 4*x*A'(x)) dx.
(3) A(x) = 1 + Integral A(x) * (1 + x*A(x)^2 - sqrt(1 - 14*x*A(x)^2 + x^2*A(x)^4) )/(8*x) dx.
(4) 0 = A(x)^4 - A(x)*(1 + x*A(x)^2)*A'(x) + 4*x*A'(x)^2.
a(n) ~ 3^(2/3) * (1240209 - 716035*sqrt(3))^(1/6) * 2^((4*n - 5)/3) * (3 + 2*sqrt(3))^n / (sqrt(Pi) * n^(5/2)). - Vaclav Kotesovec, Oct 14 2020

A303064 O.g.f. A(x) satisfies: A(x) = 1 + Integral (x/A(x)^4)' / (x/A(x)^7)' dx.

Original entry on oeis.org

1, 1, 3, 19, 181, 2121, 28035, 401199, 6076494, 96043696, 1569407902, 26338955412, 451829668906, 7894022242204, 140075248932330, 2518908613782600, 45823514062061016, 842108582257569081, 15614889082228858722, 291858857158743005901, 5494258022591894716440, 104097462455681871262881, 1983820645046435115347970, 38007365345354099879246673
Offset: 0

Views

Author

Paul D. Hanna, Apr 19 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 19*x^3 + 181*x^4 + 2121*x^5 + 28035*x^6 + 401199*x^7 + 6076494*x^8 + 96043696*x^9 + 1569407902*x^10 + ...
RELATED SERIES.
(x/A(x)^4)' / (x/A(x)^7)' = 1 + 6*x + 57*x^2 + 724*x^3 + 10605*x^4 + 168210*x^5 + 2808393*x^6 + 48611952*x^7 + 864393264*x^8 + 15694079020*x^9 + ...
which equals A'(x).
The logarithmic derivative of the g.f. begins:
A'(x)/A(x) = 1 + 5*x + 49*x^2 + 641*x^3 + 9541*x^4 + 152789*x^5 + 2567293*x^6 + 44643689*x^7 + 796602019*x^8 + 14502820745*x^9 + ...
which equals (1 + 4*x*A(x)^2 - sqrt(1 - 20*x*A(x)^2 + 16*x^2*A(x)^4))/(14*x).

Links

Crossrefs

Cf. A302701, A302704, A302705, A338163, A338187, A338188, A338193, A338194.

Programs

  • Mathematica
    nmax = 25; A = 1; Do[A = 1 + Integrate[D[x/A^4, x]/D[x/A^7, x], x] + O[x]^nmax, nmax]; CoefficientList[A, x] (* Vaclav Kotesovec, Oct 15 2020 *)
  • PARI
    {a(n) = my(A=1); for(i=1, n, A = 1 + intformal( (x/A^4)'/(x/A^7 +x*O(x^n))' ); ); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

Formula

O.g.f. A(x) satisfies:
(1) A(x) = 1 + Integral (x/A(x)^4)' / (x/A(x)^7)' dx.
(2) A(x) = 1 + Integral A(x)^3 * (A(x) - 4*x*A'(x)) / (A(x) - 7*x*A'(x)) dx.
(3) A(x) = 1 + Integral A(x) * (1 + 4*x*A(x)^2 - sqrt(1 - 20*x*A(x)^2 + 16*x^2*A(x)^4) )/(14*x) dx.
(4) 0 = A(x)^4 - A(x)*(1 + 4*x*A(x)^2)*A'(x) + 7*x*A'(x)^2.
a(n) ~ c * d^n / n^(5/2), where d = 2^(5/3)*(3/7)^(4/9)*(432893979 + 94465211*sqrt(21))^(1/9) = 21.43962319881971664190505405921680468600... and c = 0.022570265358175200394042178896826753964244... - Vaclav Kotesovec, Oct 14 2020, updated Mar 16 2024

A360347 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(2*n+3))^(n+1) for n >= 0.

Original entry on oeis.org

1, 1, 7, 105, 2366, 68776, 2390230, 95166058, 4228436480, 206090296497, 10887958126763, 618187895371965, 37479711430699245, 2414492049517400164, 164626564026042206780, 11841796830661101527618, 896184803460067359995232, 71189783172592806474895908
Offset: 0

Views

Author

Paul D. Hanna, Feb 05 2023

Keywords

Comments

Sequences with g.f. A(x,k) such that [x^n] A(x,k)^(n+1) = [x^n] (1 + x*A(x,k)^(2*n+k))^(n+1) have a rate of growth: a(n) ~ c(k) * d^n * n! * n^alpha(k), where d = 3.93464558322824528799... (independent on k) and alpha(k) = 1.1169011279372... + k*0.518500901361... - Vaclav Kotesovec, Feb 06 2023

Examples

			G.f.: A(x) = 1 + x + 7*x^2 + 105*x^3 + 2366*x^4 + 68776*x^5 + 2390230*x^6 + 95166058*x^7 + 4228436480*x^8 + 206090296497*x^9 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 8*x^2 + 127*x^3 + 2927*x^4 + 85892*x^5 + 2998264*x^6 + 119665415*x^7 + 5325877575*x^8 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(2*n+3))^(n+1) / (n+1),
as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
so that b(n) begin:
[1/1, 2/2, 24/3, 508/4, 14635/5, 515352/6, 20987848/7, 957323320/8, ...].
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1,  7,  105,  2366,  68776,  2390230,  95166058, ...];
n=1: [1, 2, 15,  224,  4991, 143754,  4962161, 196572300, ...];
n=2: [1, 3, 24,  358,  7896, 225396,  7727644, 304572936, ...];
n=3: [1, 4, 34,  508, 11103, 314192, 10699244, 419541832, ...];
n=4: [1, 5, 45,  675, 14635, 410661, 13890275, 541873525, ...];
n=5: [1, 6, 57,  860, 18516, 515352, 17314836, 671984280, ...];
n=6: [1, 7, 70, 1064, 22771, 628845, 20987848, 810313190, ...];
n=7: [1, 8, 84, 1288, 27426, 751752, 24925092, 957323320, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(2*n+3))^(n+1):
n=0: [1, 1,   3,   24,   358,   7896,   225396,   7727644, ...];
n=1: [1, 2,  11,  100,  1465,  31070,   859367,  28808972, ...];
n=2: [1, 3,  24,  253,  3780,  77994,  2089024,  68277867, ...];
n=3: [1, 4,  42,  508,  7915, 161316,  4196916, 133476480, ...];
n=4: [1, 5,  65,  890, 14635, 298981,  7602705, 235213110, ...];
n=5: [1, 6,  93, 1424, 24858, 515352, 12914214, 389369448, ...];
n=6: [1, 7, 126, 2135, 39655, 842331, 20987848, 619044602, ...];
n=7: [1, 8, 164, 3048, 60250, 1320480, 32998388, 957323320, ...]; ...
to see that the main diagonals of the tables are the same.

Links

Crossrefs

Cf. A360342, A360343, A360344, A360345, A360346.
Cf. A302704, A360235, A360237.

Programs

  • PARI
    {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(2*m+3))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(2*n+3))^(n+1) for n>=0.
(2) A(x) = Sum_{n>=0} b(n) * x^n/A(x)^n, where b(n) = [x^n] (1 + x*A(x)^(2*n+3))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alpha, where d = 3.93464558322824528799..., alpha = 2.672403832022..., c = 0.0085435225111... - Vaclav Kotesovec, Feb 06 2023

A302705 O.g.f. A(x) satisfies: A(x) = 1 + Integral (x*A(x)^4)' / (x*A(x))' dx.

Original entry on oeis.org

1, 1, 3, 11, 43, 171, 677, 2637, 10035, 37171, 134009, 472785, 1655845, 5910373, 22254507, 90625475, 396822579, 1803795507, 8151776201, 35314777505, 142395796689, 518352934225, 1625522953935, 3944383216263, 4604242439037, -17114536692099, -114353748666873, -52384917067153, 4112292989447275, 42810794269242411, 290607272326013813
Offset: 0

Views

Author

Paul D. Hanna, Apr 15 2018

Keywords

Comments

Note: if F(x) = 1 + Integral (x*F(x)^3)' / (x*F(x))' dx, then F(x) does not consist entirely of integer coefficients in its power series expansion.
Given G(x) = 1 + Integral (x*G(x)^p)' / (x*G(x)^q)' dx, for what fixed integers p and q is G(x) an integer power series?

Examples

			O.g.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 43*x^4 + 171*x^5 + 677*x^6 + 2637*x^7 + 10035*x^8 + 37171*x^9 + 134009*x^10 + 472785*x^11 + 1655845*x^12 + ...
RELATED SERIES.
(x*A(x)^4)' / (x*A(x))' = 1 + 6*x + 33*x^2 + 172*x^3 + 855*x^4 + 4062*x^5 + 18459*x^6 + 80280*x^7 + 334539*x^8 + ... + (n+1)*a(n+1)*x^n + ...
which equals A'(x).
The logarithmic derivative of the g.f. begins:
A'(x)/A(x) = 1 + 5*x + 25*x^2 + 121*x^3 + 561*x^4 + 2477*x^5 + 10361*x^6 + 40817*x^7 + 150433*x^8 + 515605*x^9 + 1646041*x^10 + ...
which equals (sqrt(1 - 4*x*A(x)^2 + 16*x^2*A(x)^4) - 1 + 4*x*A(x)^2) / (2*x).

Links

Crossrefs

Cf. A302701, A302704, A303064.

Programs

  • Maple
    ogf := RootOf(16*x^2*(4*x-1)*y^3-8*x*(3*x-1)*y^2+(9*x+1)*(3*x-1)*y+1, y)^(1/2);
gfun[seriestolist](series(ogf, x=0, 31))[]; # Mark van Hoeij, Nov 28 2024
  • PARI
    {a(n) = my(A=1); for(i=1,n, A = A = 1 + intformal( (x*A^4)'/(x*A +x*O(x^n))' );); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

O.g.f. A(x) satisfies:
(1) A(x) = 1 + Integral (x*A(x)^4)' / (x*A(x))' dx.
(2) A(x) = 1 + Integral A(x)^3 * (A(x) + 4*x*A'(x)) / (A(x) + x*A'(x)) dx.
(3) A(x) = 1 + Integral A(x) * ( sqrt(1 - 4*x*A(x)^2 + 16*x^2*A(x)^4) - 1 + 4*x*A(x)^2 ) / (2*x) dx.
(4) 0 = A(x)^4 - A(x)*(1 - 4*x*A(x)^2)*A'(x) - x*A'(x)^2.
Showing 1-4 of 4 results.

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