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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A277300 G.f. satisfies: A(x - A(x)^2) = x + 4*A(x)^2.

Original entry on oeis.org

1, 5, 60, 1000, 19970, 448160, 10926360, 283651245, 7740058300, 220046970860, 6476695275680, 196438030797880, 6117627849485360, 195082685133612800, 6355848358118392400, 211189970909192038500, 7146354688384980282000, 245970478274041025623200, 8602606263466490521359400, 305460999044315834902424200, 11003870605124169641012461600
Offset: 1

Views

Author

Paul D. Hanna, Oct 09 2016

Keywords

Examples

			G.f.: A(x) = x + 5*x^2 + 60*x^3 + 1000*x^4 + 19970*x^5 + 448160*x^6 + 10926360*x^7 + 283651245*x^8 + 7740058300*x^9 + 220046970860*x^10 +...

Links

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.
Cf. A277301, A277302, A277303, A277304, A277305, A277306, A277307, A277308, A277309.
Cf. A276364.

Programs

  • Mathematica
    m = 22; A[_] = 0;
Do[A[x_] = x + 5 A[4x/5 + A[x]/5]^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 - F^2) - 4*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( 4*x/5 + A(x)/5 )^2.
(2) A(x) = -4*x + 5 * Series_Reversion(x - A(x)^2).
(3) R(x) = -x/4 + 5/4 * Series_Reversion(x + 4*A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/5 - R(x)/5 ) ) = x/5 + 4*R(x)/5, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k) * 5^(n-k-1).

A277301 G.f. satisfies: A(x - 2*A(x)^2) = x + 3*A(x)^2.

Original entry on oeis.org

1, 5, 70, 1425, 35410, 999210, 30855820, 1020407105, 35642665050, 1302725802510, 49490450201460, 1944619121474970, 78734794663758580, 3275324221277662900, 139667810517388712600, 6093781146211490413825, 271623891311306597652650, 12353670814537544856558950, 572686428900679117724156900, 27036308383662996662940155550, 1298856469077709523772645582300
Offset: 1

Views

Author

Paul D. Hanna, Oct 09 2016

Keywords

Examples

			G.f.: A(x) = x + 5*x^2 + 70*x^3 + 1425*x^4 + 35410*x^5 + 999210*x^6 + 30855820*x^7 + 1020407105*x^8 + 35642665050*x^9 + 1302725802510*x^10 +...

Links

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.
Cf. A277300, A277302, A277303, 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 - 2*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 + 5 * A( 3*x/5 + 2*A(x)/5 )^2.
(2) A(x) = -3*x/2 + 5/2 * Series_Reversion(x - 2*A(x)^2).
(3) R(x) = -2*x/3 + 5/3 * Series_Reversion(x + 3*A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/5 - R(x)/5 ) ) = 2*x/5 + 3*R(x)/5, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k) * 2^k * 5^(n-k-1).

A277302 G.f. satisfies: A(x - 3*A(x)^2) = x + 2*A(x)^2.

Original entry on oeis.org

1, 5, 80, 1900, 55490, 1848660, 67630080, 2657251005, 110560510400, 4824793769260, 219334788340040, 10334817935549420, 502814686712631520, 25184673137026274600, 1295595210394570426800, 68326193725188929358600, 3688253200687778850553800, 203524353764195058692833200, 11468618360097679305600299400, 659345494779348103800864088800, 38644445208422874351089132287200
Offset: 1

Views

Author

Paul D. Hanna, Oct 09 2016

Keywords

Examples

			G.f.: A(x) = x + 5*x^2 + 80*x^3 + 1900*x^4 + 55490*x^5 + 1848660*x^6 + 67630080*x^7 + 2657251005*x^8 + 110560510400*x^9 + 4824793769260*x^10 +...

Links

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.
Cf. A277300, A277301, A277303, 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 - 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 + 5 * A( 2*x/5 + 3*A(x)/5 )^2.
(2) A(x) = -2*x/3 + 5/3 * Series_Reversion(x - 3*A(x)^2).
(3) R(x) = -3*x/2 + 5/2 * Series_Reversion(x + 2*A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/5 - R(x)/5 ) ) = 3*x/5 + 2*R(x)/5, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k) * 3^k * 5^(n-k-1).

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).
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