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.

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A277303 G.f. satisfies: A(x - 4*A(x)^2) = x + A(x)^2.

Original entry on oeis.org

1, 5, 90, 2425, 80630, 3065810, 128271540, 5774538945, 275743894750, 13832116773110, 723891526915820, 39323723086794730, 2208811824884144540, 127904686371063157700, 7617441454740093233000, 465691699545009287055825, 29179499379365501297165550, 1871486497257264286902367950, 122731222232573572625823907900, 8222122259910817121846641763950, 562251437460415648354364719018900
Offset: 1

Views

Author

Paul D. Hanna, Oct 09 2016

Keywords

Examples

			G.f.: A(x) = x + 5*x^2 + 90*x^3 + 2425*x^4 + 80630*x^5 + 3065810*x^6 + 128271540*x^7 + 5774538945*x^8 + 275743894750*x^9 + 13832116773110*x^10 +...

Links

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.
Cf. A277300, A277301, A277302, A277304, A277305, A277306, A277307, A277308, A277309.
Cf. A276364.

Programs

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

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + 5 * A( x/5 + 4*A(x)/5 )^2.
(2) A(x) = -x/4 + 5/4 * Series_Reversion(x - 4*A(x)^2).
(3) R(x) = -4*x + 5 * Series_Reversion(x + A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/5 - R(x)/5 ) ) = 4*x/5 + R(x)/5, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k) * 4^k * 5^(n-k-1).

A277304 G.f. satisfies: A(x - A(x)^2) = x + 5*A(x)^2.

Original entry on oeis.org

1, 6, 84, 1614, 36948, 947412, 26334072, 778107150, 24133349532, 778923367284, 26000354998920, 893459845502916, 31496296778304936, 1135911643635146712, 41820127450763818896, 1568983653501973667262, 59898843849911992994340, 2324166762372316001442540, 91565378725229449617874824, 3659689884915567083966937156, 148284110214725433666804447912
Offset: 1

Views

Author

Paul D. Hanna, Oct 09 2016

Keywords

Examples

			G.f.: A(x) = x + 6*x^2 + 84*x^3 + 1614*x^4 + 36948*x^5 + 947412*x^6 + 26334072*x^7 + 778107150*x^8 + 24133349532*x^9 + 778923367284*x^10 +...

Links

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.
Cf. A277300, A277301, A277302, A277303, A277305, A277306, A277307, A277308, A277309.
Cf. A276364.

Programs

  • PARI
    {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x - F^2) - 5*F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + 6 * A( 5*x/6 + A(x)/6 )^2.
(2) A(x) = -5*x + 6 * Series_Reversion(x - A(x)^2).
(3) R(x) = -x/5 + 6/5 * Series_Reversion(x + 5*A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/6 - R(x)/6 ) ) = x/6 + 5*R(x)/6, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k) * 6^(n-k-1).

A277305 G.f. satisfies: A(x - 5*A(x)^2) = x + A(x)^2.

Original entry on oeis.org

1, 6, 132, 4350, 176964, 8235252, 421814232, 23252672574, 1359954622860, 83572511671092, 5359130778285096, 356786692299782916, 24565803644793789192, 1744056102774572824920, 127369971591949093219920, 9550397045409732902387790, 734084078724419876468356500, 57766855968717521513179054860, 4648888743682938087701732224680
Offset: 1

Views

Author

Paul D. Hanna, Oct 09 2016

Keywords

Examples

			G.f.: A(x) = x + 6*x^2 + 132*x^3 + 4350*x^4 + 176964*x^5 + 8235252*x^6 + 421814232*x^7 + 23252672574*x^8 + 1359954622860*x^9 + 83572511671092*x^10 +...

Links

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.
Cf. A277300, A277301, A277302, A277303, A277304, A277306, A277307, A277308, A277309.
Cf. A276364.

Programs

  • PARI
    {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x - 5*F^2) - F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + 6 * A( x/6 + 5*A(x)/6 )^2.
(2) A(x) = -x/5 + 6/5 * Series_Reversion(x - 5*A(x)^2).
(3) R(x) = -5*x + 6 * Series_Reversion(x + A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/6 - R(x)/6 ) ) = 5*x/6 + R(x)/6, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k) * 5^k * 6^(n-k-1).

A277306 G.f. satisfies: A(x + A(x)^2) = x + 2*A(x)^2.

Original entry on oeis.org

1, 1, 0, -4, 2, 52, -96, -975, 4240, 18460, -183448, -101716, 7373216, -23650520, -230147920, 2198499720, 664806792, -124144328784, 703989911368, 3189500786336, -68800373946656, 284782780974128, 2913071885553608, -47063844278787824, 170357147598919640, 2621783446017272624, -41775596442709927664, 166446909354828214608
Offset: 1

Views

Author

Paul D. Hanna, Oct 09 2016

Keywords

Examples

			G.f.: A(x) = x + x^2 - 4*x^4 + 2*x^5 + 52*x^6 - 96*x^7 - 975*x^8 + 4240*x^9 + 18460*x^10 - 183448*x^11 - 101716*x^12 + 7373216*x^13 - 23650520*x^14 - 230147920*x^15 + 2198499720*x^16 + 664806792*x^17 - 124144328784*x^18 + 703989911368*x^19 + 3189500786336*x^20 +...
such that
A(x + A(x)^2) = x + 2*A(x)^2
also,
A(x) = x + A( 2*x - A(x) )^2.
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + x^4 - 8*x^5 - 4*x^6 + 108*x^7 - 72*x^8 - 2158*x^9 + 6118*x^10 + 46376*x^11 - 319856*x^12 - 618132*x^13 + 14320096*x^14 - 30385024*x^15 - 505460559*x^16 + 3846420096*x^17 + 5951934200*x^18 - 243911854368*x^19 + 1136290742936*x^20 +...
A(x + A(x)^2) = x + 2*x^2 + 4*x^3 + 2*x^4 - 16*x^5 - 8*x^6 + 216*x^7 - 144*x^8 - 4316*x^9 + 12236*x^10 + 92752*x^11 - 639712*x^12 +...
which equals x + 2*A(x)^2.
Series_Reversion(A(x)) = x - x^2 + 2*x^3 - x^4 - 12*x^5 + 32*x^6 + 156*x^7 - 1140*x^8 - 1178*x^9 + 41270*x^10 - 105480*x^11 - 1274828*x^12 + 10307292*x^13 + 13297704*x^14 - 609624768*x^15 + 2614447647*x^16 + 21136068780*x^17 - 300421913212*x^18 + 590894313656*x^19 + 17309654827168*x^20 +...
which equals 2*x - Series_Reversion(x + 2*A(x)^2).

Links

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.
Cf. A277300, A277301, A277302, A277303, A277304, A277305, A277307, A277308, A277309, A277310, A277311.
Cf. A276364.

Programs

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

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + A( 2*x - A(x) )^2.
(2) A(x) = 2*x - Series_Reversion(x + A(x)^2).
(3) R(x) = x/2 + 1/2 * Series_Reversion(x + 2*A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x - R(x) ) ) = -x + 2*R(x), where R(A(x)) = x.
(5) A(x) = x + Sum_{n>=1} (-1)^(n-1) * d^(n-1)/dx^(n-1) A(x)^(2*n) / n!.
a(n) = Sum_{k=0..n-1} (-1)^k * A277295(n,k).

A277307 G.f. satisfies: A(x - 3*A(x)^2) = x - 2*A(x)^2.

Original entry on oeis.org

1, 1, 8, 92, 1298, 20988, 375120, 7252065, 149534312, 3256987724, 74418884792, 1774657501252, 43995940957120, 1130453689908568, 30031716838365552, 823263454676130312, 23249951990747403528, 675517165191231019920, 20168579968950108809736, 618158189347428262782816, 19432224179107494743506272, 626034612821085407187912624
Offset: 1

Views

Author

Paul D. Hanna, Oct 09 2016

Keywords

Examples

			G.f.: A(x) = x + x^2 + 8*x^3 + 92*x^4 + 1298*x^5 + 20988*x^6 + 375120*x^7 + 7252065*x^8 + 149534312*x^9 + 3256987724*x^10 +...
such that A(x - 3*A(x)^2) = x - 2*A(x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 17*x^4 + 200*x^5 + 2844*x^6 + 46044*x^7 + 821448*x^8 + 15829010*x^9 + 325121270*x^10 + 7052584040*x^11 + 160492981648*x^12 + 3812351286940*x^13 + 94164503583424*x^14 + 2411159638210752*x^15 + 63849498902714289*x^16 +...
A(x - 3*A(x)^2) = x - 2*x^2 - 4*x^3 - 34*x^4 - 400*x^5 - 5688*x^6 - 92088*x^7 - 1642896*x^8 - 31658020*x^9 - 650242540*x^10 +...
which equals x - 2*A(x)^2.
Series_Reversion(x - 3*A(x)^2) = x + 3*x^2 + 24*x^3 + 276*x^4 + 3894*x^5 + 62964*x^6 + 1125360*x^7 + 21756195*x^8 + 448602936*x^9 + 9770963172*x^10 +...
which equals -2*x + 3*A(x).
A( 3*A(x) - 2*x ) = x + 4*x^2 + 38*x^3 + 497*x^4 + 7784*x^5 + 137538*x^6 + 2656584*x^7 + 55045728*x^8 + 1208709044*x^9 + 27891950516*x^10 +...
which equals sqrt( A(x) - x ).

Links

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.
Cf. A277300, A277301, A277302, A277303, A277304, A277305, A277306, A277308, A277309.
Cf. A276364.

Programs

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

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + A( 3*A(x) - 2*x )^2.
(2) A(x) = 2*x/3 + 1/3 * Series_Reversion(x - 3*A(x)^2).
(3) R(x) = 3*x/2 - 1/2 * Series_Reversion(x - 2*A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x - R(x) ) ) = 3*x - 2*R(x), where R(A(x)) = x.
(5) A(x) = x + Sum_{n>=1} 3^(n-1) * d^(n-1)/dx^(n-1) A(x)^(2*n) / n!.
a(n) = Sum_{k=0..n-1} A277295(n,k) * 3^k.

A277308 G.f. satisfies: A(x - 3*A(x)^2) = x - A(x)^2.

Original entry on oeis.org

1, 2, 20, 298, 5492, 116124, 2710776, 68308170, 1831522940, 51744512380, 1529687560328, 47075470016012, 1502258036769256, 49560341916549320, 1686236991420431760, 59054595629732284890, 2125432920387784135812, 78509698415432235272292, 2972996232264052816975752, 115303660044380692013332428
Offset: 1

Views

Author

Paul D. Hanna, Oct 09 2016

Keywords

Examples

			G.f.: A(x) = x + 2*x^2 + 20*x^3 + 298*x^4 + 5492*x^5 + 116124*x^6 + 2710776*x^7 + 68308170*x^8 + 1831522940*x^9 + 51744512380*x^10 +...

Links

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.
Cf. A277300, A277301, A277302, A277303, A277304, A277305, A277306, A277307, A277309.
Cf. A276364.

Programs

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

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + 2 * A( 3*A(x)/2 - x/2 )^2.
(2) A(x) = x/3 + 2/3 * Series_Reversion(x - 3*A(x)^2).
(3) R(x) = 3*x - 2 * Series_Reversion(x - A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/2 - R(x)/2 ) ) = 3*x/2 - R(x)/2, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k) * 3^k * 2^(n-k-1).

A277309 G.f. satisfies: A(x - 5*A(x)^2) = x - 3*A(x)^2.

Original entry on oeis.org

1, 2, 28, 570, 14284, 410604, 13046728, 448252682, 16417945620, 634848045084, 25737059674104, 1088311917852828, 47813839403065432, 2175881570186952520, 102316326149365110320, 4961686220242926811690, 247733650768933667153660, 12718117037478356041212500, 670565414769224589112024760, 36274908884974158393988101900, 2011581759381610503724213971960
Offset: 1

Views

Author

Paul D. Hanna, Oct 09 2016

Keywords

Examples

			G.f.: A(x) = x + 2*x^2 + 28*x^3 + 570*x^4 + 14284*x^5 + 410604*x^6 + 13046728*x^7 + 448252682*x^8 + 16417945620*x^9 + 634848045084*x^10 +...

Links

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.
Cf. A277300, A277301, A277302, A277303, A277304, A277305, A277306, A277307, A277308.
Cf. A276364.

Programs

  • PARI
    {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x-5*F^2) + 3*F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + 2 * A( 5*A(x)/2 - 3*x/2 )^2.
(2) A(x) = 3*x/5 + 2/5 * Series_Reversion(x - 5*A(x)^2).
(3) R(x) = 5*x/3 - 2/3 * Series_Reversion(x - 3*A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/2 - R(x)/2 ) ) = 5*x/2 - 3*R(x)/2, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k) * 5^k * 2^(n-k-1).

A276358 G.f. A(x) satisfies: A(x - x*A(x)) = x + x*A(x).

Original entry on oeis.org

1, 2, 8, 46, 324, 2608, 23136, 221370, 2252872, 24153284, 270922880, 3163154736, 38291322000, 479133266432, 6181998751808, 82084129578414, 1119798740473788, 15674024566862424, 224843628257016920, 3302256609111585300, 49613275311027132672, 761926428688868584400, 11952618573953745931536, 191418290850831848697272, 3127755564602007721663352, 52118116918762815035493760, 885205781290692080951844800, 15318116453244882343710519680, 269953482313408263924956600000
Offset: 1

Views

Author

Paul D. Hanna, Sep 04 2016

Keywords

Examples

			G.f.: A(x) = x + 2*x^2 + 8*x^3 + 46*x^4 + 324*x^5 + 2608*x^6 + 23136*x^7 + 221370*x^8 + 2252872*x^9 + 24153284*x^10 + 270922880*x^11 + 3163154736*x^12 +...
such that A(x - x*A(x)) = x + x*A(x).
RELATED SERIES.
A(x - x*A(x)) = x + x^2 + 2*x^3 + 8*x^4 + 46*x^5 + 324*x^6 + 2608*x^7 +...
which equals x + x*A(x).
Series_Reversion( x - x*A(x) ) = x + x^2 + 4*x^3 + 23*x^4 + 162*x^5 + 1304*x^6 + 11568*x^7 + 110685*x^8 + 1126436*x^9 + 12076642*x^10 + 135461440*x^11 + 1581577368*x^12 +...
which equals (x + A(x))/2.
A( (x + A(x))/2 ) = x + 3*x^2 + 16*x^3 + 111*x^4 + 898*x^5 + 8068*x^6 + 78400*x^7 + 810875*x^8 + 8832804*x^9 + 100592970*x^10 + 1191393144*x^11 + 14616198024*x^12 +...
which equals (A(x) - x)/(A(x) + x).

Links

Crossrefs

Cf. A275765.

Programs

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

Formula

G.f. A(x) satisfies:
(1) A(x) = 2 * Series_Reversion( x - x*A(x) ) - x.
(2) A(x) = x * (1 + A(B(x))) / (1 - A(B(x))), where B(x) = (x + A(x))/2.
(3) A( (x + A(x))/2 ) = (A(x) - x) / (A(x) + x).

A277310 G.f. satisfies: A(x - 4*A(x)^2) = x - 3*A(x)^2.

Original entry on oeis.org

1, 1, 10, 141, 2422, 47562, 1031764, 24214405, 606444990, 16055089470, 446238074892, 12955112773554, 391332183548956, 12261884937532340, 397576302315045800, 13313017677172350965, 459635990935574444942, 16339309997761322057206, 597340515437542895494748, 22435278085988347895795526, 864900964565994975048855444, 34195693888939483596581262668, 1385553440866978431053220575128
Offset: 1

Views

Author

Paul D. Hanna, Oct 12 2016

Keywords

Examples

			G.f.: A(x) = x + x^2 + 10*x^3 + 141*x^4 + 2422*x^5 + 47562*x^6 + 1031764*x^7 + 24214405*x^8 + 606444990*x^9 + 16055089470*x^10 +...
such that A(x - 4*A(x)^2) = x - 3*A(x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 21*x^4 + 302*x^5 + 5226*x^6 + 102788*x^7 + 2226973*x^8 + 52126582*x^9 + 1301232638*x^10 + 34328704796*x^11 + 950803699394*x^12 + 27510261070028*x^13 + 828332416917876*x^14 + 25876801064095496*x^15 + 836682915170627501*x^16 +...
A(x - 4*A(x)^2) = x - 3*x^2 - 6*x^3 - 63*x^4 - 906*x^5 - 15678*x^6 - 308364*x^7 - 6680919*x^8 - 156379746*x^9 - 3903697914*x^10 +...
which equals x - 3*A(x)^2.
Series_Reversion(x - 4*A(x)^2) = x + 4*x^2 + 40*x^3 + 564*x^4 + 9688*x^5 + 190248*x^6 + 4127056*x^7 + 96857620*x^8 + 2425779960*x^9 + 64220357880*x^10 +...
which equals -3*x + 4*A(x).
A( 4*A(x) - 3*x ) = x + 5*x^2 + 58*x^3 + 921*x^4 + 17494*x^5 + 374994*x^6 + 8793460*x^7 + 221393569*x^8 + 5912166718*x^9 + 166058455158*x^10 + 4876311925036*x^11 + 149037482367530*x^12 + 4724877954111836*x^13 + 154959634972646340*x^14 + 5246331138228520168*x^15 +...
which equals  sqrt( A(x) - x ).

Links

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.
Cf. A277300, A277301, A277302, A277303, A277304, A277305, A277306, A277307, A277308, A277309.

Programs

  • PARI
    {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x-4*F^2) + 3*F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + A( 4*A(x) - 3*x )^2.
(2) A(x) = 3*x/4 + 1/4 * Series_Reversion(x - 4*A(x)^2).
(3) R(x) = 4*x/3 - 1/3 * Series_Reversion(x - 3*A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x - R(x) ) ) = 4*x - 3*R(x), where R(A(x)) = x.
(5) A(x) = x + Sum_{n>=1} 4^(n-1) * d^(n-1)/dx^(n-1) A(x)^(2*n) / n!.
a(n) = Sum_{k=0..n-1} A277295(n,k) * 4^k.

A276365 G.f. A(x) satisfies: A(x - 2*A(x)^2) = x - A(x)^2.

Original entry on oeis.org

1, 1, 6, 53, 578, 7234, 100044, 1495125, 23802346, 399740086, 7032766196, 128952474242, 2454645604820, 48359400068836, 983683769369624, 20618782389897333, 444636132851851386, 9851377271964349038, 223998085060636514020, 5221799494107885481430, 124695762315403816775932, 3047952254964607540099676, 76206565881709345978097960, 1947752912315470845518308642, 50860833685759573411702643972
Offset: 1

Views

Author

Paul D. Hanna, Sep 01 2016

Keywords

Examples

			G.f.: A(x) = x + x^2 + 6*x^3 + 53*x^4 + 578*x^5 + 7234*x^6 + 100044*x^7 + 1495125*x^8 + 23802346*x^9 + 399740086*x^10 + 7032766196*x^11 +...
such that A(x - 2*A(x)^2) = x - A(x)^2.
RELATED SERIES.
Note that Series_Reversion(x - 2*A(x)^2) = 2*A(x) - x, which begins:
Series_Reversion(x - 2*A(x)^2) = x + 2*x^2 + 12*x^3 + 106*x^4 + 1156*x^5 + 14468*x^6 + 200088*x^7 + 2990250*x^8 + 47604692*x^9 + 799480172*x^10 +...
Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
R(x) = x - x^2 - 4*x^3 - 28*x^4 - 264*x^5 - 2992*x^6 - 38496*x^7 - 544464*x^8 - 8298080*x^9 - 134500672*x^10 - 2297361024*x^11 +...
then Series_Reversion(x - A(x)^2) = 2*x - R(x), and
R(x) = x - G(x)^2, where G(x) = x + 2*x^2 + 12*x^3 + 108*x^4 + 1208*x^5 + 15536*x^6 + 220832*x^7 + 3390480*x^8 + ... + A177409(n)*x^n + ...
Also, sqrt(A(x) - x) = A(2*A(x) - x), which begins:
sqrt(A(x) - x) = x + 3*x^2 + 22*x^3 + 223*x^4 + 2706*x^5 + 36998*x^6 + 552172*x^7 + 8827263*x^8 + 149328698*x^9 + 2650946274*x^10 + ...

Links

Crossrefs

Cf. A277295, A213591, A275765, A177409.

Programs

  • Mathematica
    m = 26; A[_] = 0;
Do[A[x_] = x + A[2 A[x] - x]^2 + O[x]^m // Normal, {m}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Sep 30 2019 *)
  • PARI
    {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x - 2*F^2) + F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + A( 2*A(x) - x )^2.
(2) 2*A(x) = x + Series_Reversion(x - 2*A(x)^2).
(3) R(x) = 2*x - Series_Reversion(x - A(x)^2), where R(A(x)) = x.
(4) R( (x - R(x))^(1/2) ) = 2*x - R(x), where R(A(x)) = x.
(5) A(x) = x + Sum_{n>=1} 2^(n-1) * d^(n-1)/dx^(n-1) A(x)^(2*n) / n!.
(6) A(x) = x + G(A(x))^2, where G(x) = sqrt(x - R(x)) is the g.f. of A177409, and R(A(x)) = x. - Paul D. Hanna, Nov 18 2022
a(n) = Sum_{k=0..n-1} A277295(n,k)*2^k.
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