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.

A271960 G.f. A(x) satisfies: A(x)^2 = A( (x + 2*A(x)^2)^2 ).

Original entry on oeis.org

1, 2, 9, 50, 312, 2086, 14613, 105864, 786627, 5962110, 45914544, 358247214, 2825957294, 22499804332, 180573770279, 1459277489372, 11864714598122, 96985441764430, 796580710229999, 6570692234061404, 54408498662798180, 452104483291381134, 3768693666865385520, 31506775300298343840, 264103426399414754616, 2219265880819182687882, 18690831189839369444283, 157746446435747834724764, 1333944139058301773582424
Offset: 1

Views

Author

Paul D. Hanna, Apr 23 2016

Keywords

Examples

			G.f.: A(x) = x + 2*x^2 + 9*x^3 + 50*x^4 + 312*x^5 + 2086*x^6 + 14613*x^7 + 105864*x^8 + 786627*x^9 + 5962110*x^10 + 45914544*x^11 + 358247214*x^12 +...
where A(x)^2 = A( (x + 2*A(x)^2)^2 ).
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 22*x^4 + 136*x^5 + 905*x^6 + 6320*x^7 + 45686*x^8 + 338928*x^9 + 2565688*x^10 + 19739244*x^11 + 153893122*x^12 +...
(x + 2*A(x)^2)^2 = x^2 + 4*x^3 + 20*x^4 + 120*x^5 + 784*x^6 + 5412*x^7 + 38808*x^8 + 286200*x^9 + 2156704*x^10 + 16533088*x^11 + 128521172*x^12 +...
sqrt(x*A(x)) = x + x^2 + 4*x^3 + 21*x^4 + 127*x^5 + 832*x^6 + 5746*x^7 + 41191*x^8 + 303602*x^9 + 2286359*x^10 + 17515640*x^11 + 136074960*x^12 + 1069490964*x^13 + 8488634979*x^14 + 67943128844*x^15 + 547784144486*x^16 +...
Let B(x) be the series reversion of g.f. A(x), so that A(B(x)) = x, then
B(x) = x - 2*x^2 - x^3 - x^5 - x^7 - 2*x^9 - 3*x^11 - 6*x^13 - 11*x^15 - 23*x^17 - 46*x^19 - 98*x^21 - 207*x^23 - 451*x^25 +...+ -A001190(n)*x^(2*n+1) +...
such that B(x) = x - 2*x^2 - x*G(x^2), where G(x) = x + (1/2)*(G(x)^2 + G(x^2)).
Let C(x) = series reversion of sqrt(x*A(x)), so that C(x)*A(C(x)) = x^2, then
A(C(x)) = F(x) = x + x^2 + 3*x^3 + 11*x^4 + 46*x^5 + 206*x^6 + 968*x^7 + 4706*x^8 + 23475*x^9 + 119473*x^10 +...+ A271959(n)*x^n +...
where F(x)^2 = F( x^2 + 2*F(x)^3 ).
SPECIFIC VALUES.
A(1/9) = 0.193716899378316704135816541113258730084910624425207...
where A(1/9)^2 = A( (1/9 + 2*A(1/9)^2)^2 ).
A(1/10) = 0.14546429517891287614768130046564115776024954490631...
where A(1/10)^2 = A( (1/10 + 2*A(1/10)^2)^2 ).
A(1/11) = 0.12312262826542819891757177602341377693863412904549...
where A(1/11)^2 = A( (1/11 + 2*A(1/11)^2)^2 ).
A(1/12) = 0.10787967285688466984977060963500355819842649026285...
where A(1/12)^2 = A( (1/12 + 2*A(1/12)^2)^2 ).

Links

Crossrefs

Cf. A271959, A001190.

Programs

  • PARI
    {a(n) = my(A=x+x^2, X=x+x*O(x^n)); for(i=1, n, A = subst(A, x, (X + 2*A^2)^2 )^(1/2) ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A(x - 2*x^2 - x*G(x^2)) = x, where G(x) = x + (1/2)*(G(x)^2 + G(x^2)) is the g.f. of the Wedderburn-Etherington numbers (A001190).
(2) A(x) = F( sqrt(x*A(x)) ) where F(x)^2 = F( x^2 + 2*F(x)^3 ) and F(x) is the g.f. of A271959.
a(n) ~ c * d^n / n^(3/2), where d = 8.9175668047902516038346068989... and c = 0.056993950617012713508863076... . - Vaclav Kotesovec, May 03 2016

A374566 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2 + 2*(1+x)*A(x)^3 ).

Original entry on oeis.org

1, 1, 4, 16, 76, 381, 2010, 10955, 61265, 349472, 2025632, 11896039, 70632739, 423300099, 2557174039, 15555534859, 95202925651, 585799778042, 3621806301246, 22488577587970, 140176525844646, 876813040040057, 5501997007343589, 34625517090342459, 218489435424317825, 1382072993052136903
Offset: 1

Views

Author

Paul D. Hanna, Aug 12 2024

Keywords

Comments

Compare to: C(x)^2 = C( x^2 - 2*C(x)^3 ), where C(x) = x - C(x)^2.

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 16*x^4 + 76*x^5 + 381*x^6 + 2010*x^7 + 10955*x^8 + 61265*x^9 + 349472*x^10 + ...
where A(x)^2 = A( x^2 + 2*(1+x)*A(x)^3 ).
RELATED SERIES.
Let G(x) be the g.f. of the Wedderburn-Etherington numbers, then
A( x - x^3 - x*G(x) ) = x, where G(x) = x + (1/2)*(G(x)^2 + G(x^2)) begins
G(x) = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 11*x^7 + 23*x^8 + 46*x^9 + 98*x^10 + 207*x^11 + 451*x^12 + 983*x^13 + ... + A001190(n)*x^n + ...
A(x)^2 = x^2 + 2*x^3 + 9*x^4 + 40*x^5 + 200*x^6 + 1042*x^7 + 5646*x^8 + 31410*x^9 + 178488*x^10 + 1031346*x^11 + 6041569*x^12 + ...
A(x)^3 = x^3 + 3*x^4 + 15*x^5 + 73*x^6 + 384*x^7 + 2079*x^8 + 11584*x^9 + 65868*x^10 + 380859*x^11 + 2232199*x^12 + 13231686*x^13 + ...
x^2 + 2*(1+x)*A(x)^3 = x^2 + 2*x^3 + 8*x^4 + 36*x^5 + 176*x^6 + 914*x^7 + 4926*x^8 + 27326*x^9 + 154904*x^10 + 893454*x^11 + ...
SPECIFIC VALUES.
A(t) = 1/4 at t = 0.14894182268166520428651100246692394784806895864208130...
where 1/16 = A( t^2 + (1 + t)/32 ).
A(t) = 1/5 at t = 0.14144303881517477480553509807420585604076735607834555...
where 1/25 = A( t^2 + 2*(1 + t)/125 ).
A(1/7) = 0.204913420188897006601259679664181034021504614738141...
where A(1/7)^2 = A( 1/7^2 + (16/7)*A(1/7)^3 ).
A(1/8) = 0.159462997675623738517233384699423553894402512640906...
where A(1/8)^2 = A( 1/8^2 + (18/8)*A(1/8)^3 ).
A(1/9) = 0.134511672187656270338825814076702307725993232871545...
where A(1/9)^2 = A( 1/9^2 + (20/9)*A(1/9)^3 ).
A(1/10) = 0.117197825788422212715965141990212003609448403429416...
where A(1/10)^2 = A( 1/10^2 + (22/10)*A(1/10)^3 ).

Links

Crossrefs

Cf. A374567, A271959, A001190.

Programs

  • PARI
    {a(n) = my(A=[0,1],Ax=x); for(i=1,n, A = concat(A,0); Ax=Ser(A);
A[#A] = (1/2)*polcoeff( subst(Ax,x, x^2 + 2*(1+x)*Ax^3) - Ax^2,#A) );A[n+1]}
for(n=1,30, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^2 = A( x^2 + 2*(1+x)*A(x)^3 ).
(2) x = A( x - x^3 - x*G(x) ), where G(x) = x + (1/2)*(G(x)^2 + G(x^2)) is the g.f. of A001190, the Wedderburn-Etherington numbers.
(3) x^2 = A( x^2*(1 - G(x))^2 + 2*x^3 - x^6 ), where G(x) is the g.f. of A001190.
(4) x = A( x*sqrt(1 - 2*x - G(x^2)) - x^3 ), where G(x) is the g.f. of A001190.

A374567 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2 + 2*x*A(x)^2 + 2*A(x)^3 ).

Original entry on oeis.org

1, 2, 9, 51, 325, 2222, 15926, 118085, 898217, 6970053, 54960439, 439112322, 3547096393, 28921270773, 237704587991, 1967321998468, 16381661824340, 137144132047520, 1153655788549216, 9746264972136632, 82656795697147384, 703459159019830315, 6005956718852682504, 51426768620398474939
Offset: 1

Views

Author

Paul D. Hanna, Aug 13 2024

Keywords

Comments

Compare to: C(x)^2 = C( x^2 - 2*C(x)^3 ), where C(x) = x - C(x)^2.

Examples

			G.f.: A(x) = x + 2*x^2 + 9*x^3 + 51*x^4 + 325*x^5 + 2222*x^6 + 15926*x^7 + 118085*x^8 + 898217*x^9 + 6970053*x^10 + ...
where A(x)^2 = A( x^2 + 2*x*A(x)^2 + 2*A(x)^3 ).
RELATED SERIES.
Let G(x) be the g.f. of the Wedderburn-Etherington numbers, then
A( x - x^2 - x*G(x) ) = x, where G(x) = x + (1/2)*(G(x)^2 + G(x^2)) begins
G(x) = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 11*x^7 + 23*x^8 + 46*x^9 + 98*x^10 + 207*x^11 + 451*x^12 + 983*x^13 + ... + A001190(n)*x^n + ...
A(x)^2 = x^2 + 4*x^3 + 22*x^4 + 138*x^5 + 935*x^6 + 6662*x^7 + 49191*x^8 + 373020*x^9 + 2887711*x^10 + 22727256*x^11 + ...
A(x)^3 = x^3 + 6*x^4 + 39*x^5 + 269*x^6 + 1938*x^7 + 14418*x^8 + 109932*x^9 + 854568*x^10 + 6747672*x^11 + ...
SPECIFIC VALUES.
A(t) = 1/5 at t = 0.1094430388151747748055350980742058560407673560783455...
where 1/25 = A( t^2 + 2*t/25 + 2/125 ).
A(t) = 1/6 at t = 0.1053569291935061227625330002451062383852684202941979...
where 1/36 = A( t^2 + t/18 + 1/108 ).
A(1/10) = 0.1471263013840628871589336795118257882025452972700045...
where A(1/10)^2 = A( 1/10^2 + (2/10)*A(1/10)^2 + 2*A(1/10)^3 ).
A(1/11) = 0.1237258078822115859596611191115342221543387518134407...
A(1/12) = 0.1081759735424269717469930892718654709953905803313352...

Links

Crossrefs

Cf. A374566, A271959, A001190.

Programs

  • PARI
    {a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A = concat(A, 0); Ax=Ser(A);
A[#A] = (1/2)*polcoeff( subst(Ax, x, x^2 + 2*x*Ax^2 + 2*Ax^3) - Ax^2, #A) ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^2 = A( x^2 + 2*x*A(x)^2 + 2*A(x)^3 ).
(2) x = A( x - x^2 - x*G(x) ), where G(x) = x + (1/2)*(G(x)^2 + G(x^2)) is the g.f. of A001190, the Wedderburn-Etherington numbers.
(3) x^2 = A( x^2*(1 - G(x))^2 + 2*x^3 - x^4 ), where G(x) is the g.f. of A001190.
(4) x = A( x*sqrt(1 - 2*x - G(x^2)) - x^2 ), where G(x) is the g.f. of A001190.

A378247 G.f. A(x) satisfies A(x)^2 = A( x^2 + 2*x*A(x)^2 + 2*A(x)^4 ).

Original entry on oeis.org

1, 1, 3, 10, 39, 161, 699, 3135, 14427, 67716, 322959, 1560585, 7624007, 37593476, 186856061, 935214523, 4709265692, 23841104525, 121275951719, 619558165489, 3177346503440, 16351749778167, 84419824808865, 437105510426235, 2269266695980449, 11810014285000263, 61602685079710638
Offset: 1

Views

Author

Paul D. Hanna, Nov 20 2024

Keywords

Comments

Compare to C(x)^2 = C( x^2 + 2*x*C(x)^2 ) where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

Examples

			G.f.: A(x) = x + x^2 + 3*x^3 + 10*x^4 + 39*x^5 + 161*x^6 + 699*x^7 + 3135*x^8 + 14427*x^9 + 67716*x^10 + 322959*x^11 + 1560585*x^12 + ...
where A(x)^2 = A( x^2 + 2*x*A(x)^2 + 2*A(x)^4 ).
RELATED SERIES.
Let G(x) be the g.f. of the Wedderburn-Etherington numbers, then
A( x - x^2 - x*G(x^2) ) = x, where G(x) = x + (1/2)*(G(x)^2 + G(x^2)) begins
G(x) = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 11*x^7 + 23*x^8 + 46*x^9 + 98*x^10 + 207*x^11 + 451*x^12 + 983*x^13 + 2179*x^14 + ... + A001190(n)*x^n + ...
Let B(x) be the series reversion of g.f. A(x) so that B(A(x)) = x, then
B(x) = x - x^2 - x^3 - x^5 - x^7 - 2*x^9 - 3*x^11 - 6*x^13 - 11*x^15 - 23*x^17 - 46*x^19 - 98*x^21 - 207*x^23 + ...
where B(x) = x - x^2 - x*G(x^2).
SPECIFIC VALUES.
A(t) = 1/3 at t = 0.1804894059505127351310871071614416167035910065610113327...
  where 1/9 = A( t^2 + 2*t/9 + 2/81 ).
A(t) = 1/4 at t = 0.1708289565101545485579649480920097855916395263217351536...
  where 1/16 = A( t^2 + t/8 + 1/128 ).
A(t) = 1/5 at t = 0.1516661092515691718015998101146470241027491658579501286...
  where 1/25 = A( t^2 + 2*t/25 + 2/625 ).
A(t) = 1/6 at t = 0.1341268789797555579297424694390747929782019601987848246...
  where 1/36 = A( t^2 + t/18 + 1/648 ).
A(1/6) = 0.2368314953172156547771056118501694080205525703518284958...
A(1/7) = 0.1824082884402163049324182135107985537409785918465705698...
A(1/8) = 0.1515179821748020682616541846638756124979071552818869937...
A(1/9) = 0.1303577455916869424988611259176631850931169441135101392...
A(1/10) = 0.1146797533131163787803333792504789207692884367435306666...

Links

Crossrefs

Cf. A001190, A374567, A271959, A271960, A374566.

Programs

  • PARI
    {a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A = concat(A, 0); Ax=Ser(A);
A[#A] = (1/2)*polcoeff( subst(Ax, x, x^2 + 2*x*Ax^2 + 2*Ax^4) - Ax^2, #A) ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas in which G(x) = x + (1/2)*(G(x)^2 + G(x^2)) is the g.f. of A001190, the Wedderburn-Etherington numbers.
(1) A(x)^2 = A( x^2 + 2*x*A(x)^2 + 2*A(x)^4 ).
(2) x = A( x - x^2 - x*G(x^2) ).
(3) x = A( x + x^2 - 2*x*G(x) + x*G(x)^2 ).
(4) x = A( x*sqrt(1 - 2*x^2 - G(x^4)) - x^2 ).
(5) x^2 = A( x^2*((1 - G(x))^2 + 2*x)^2 + x^4 ).
(6) G(A(x)) = 1 - sqrt(x/A(x) - A(x)).
Showing 1-4 of 4 results.

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