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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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).

A153851 Nonzero coefficients of the g.f. that satisfies: A(x) = x + A(A(x))^3.

Original entry on oeis.org

1, 1, 6, 57, 683, 9474, 145815, 2430393, 43202448, 810629805, 15938815794, 326653743510, 6949638584208, 153009877730525, 3477623225388063, 81429702521625843, 1961136442605508341, 48513571089988199157
Offset: 1

Views

Author

Paul D. Hanna, Jan 21 2009

Keywords

Examples

			G.f.: A(x) = x + x^3 + 6*x^5 + 57*x^7 + 683*x^9 + 9474*x^11 +...
A(x - A(x)^3) = x where
A(x)^3 = x^3 + 3*x^5 + 21*x^7 + 208*x^9 + 2517*x^11 + 34851*x^13 +...
SYSTEM OF RELATED FUNCTIONS.
A = A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations:
A = 1 + x^2*B^3;
B = A + x^2*C^3;
C = B + x^2*D^3;
D = C + x^2*E^3;
E = D + x^2*F^3; ...
where the functions xB, xC, xD, etc., are successive iterations of A(x):
x*A = A(x),
x*B = A(A(x)) = g.f. of A153852,
x*C = A(A(A(x))) = g.f. of A153853,
x*D = A(A(A(A(x)))) = g.f. of A153854, etc.
The nonzero coefficients of these functions begin:
A:[1, 1, 6, 57, 683, 9474, 145815, 2430393, 43202448,...];
B:[1, 2, 15, 165, 2213, 33693, 561867, 10053141, 190489374,...];
C:[1, 3, 27, 339, 5067, 84738, 1536867, 29687772, 603835479,...];
D:[1, 4, 42, 594, 9827, 179928, 3545637, 73988631, 1618178067,...];
E:[1, 5, 60, 945, 17180, 342765, 7316178, 164606166, 3866962617,...];
F:[1, 6, 81, 1407, 27918, 603879, 13907133, 336334443, 8466942393,...];
G:[1, 7, 105, 1995, 42938, 1001973, 24795645, 642380025, 17278647147,...];
H:[1, 8, 132, 2724, 63242, 1584768, 41975610, 1160887350, 33260962995,..]; ...
The main diagonal in the above table is A153850.

Links

Crossrefs

Cf. A153852, A153853, A153854, A153850; variants: A139702, A213591.

Programs

  • PARI
    {a(n)=local(A=x+x^2); for(i=0, n, A=serreverse(x-subst(A^3, x, x+x^2*O(x^(2*n))))) ; polcoeff(A, 2*n-1)}

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1) satisfies:
(1) A(x) = Series_Reversion( x - A(x)^3 ).
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(3*n) / n!. - Paul D. Hanna, Sep 07 2020
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(3*n)/x / n! ). - Paul D. Hanna, Sep 07 2020
(4) x = A(A( x-x^3 - A(x)^3 )). - Paul D. Hanna, Sep 07 2020

A190761 Expansion of g.f. A(x) satisfying A(x) = x + A(A(x))^2 - A(A(x))^3.

Original entry on oeis.org

1, 1, 3, 14, 84, 592, 4670, 40108, 368670, 3586321, 36632763, 390694000, 4332131804, 49777965585, 591173511887, 7241437905916, 91331043654080, 1184322726542850, 15770586926609276, 215423253906689779, 3015794930248824111, 43233248160139146114
Offset: 1

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Author

Paul D. Hanna, May 18 2011

Keywords

Comments

Compare to a formula for a g.f. of the Catalan numbers (A000108):
C(x) = x + C(x)*C(C(x)) - C(x)*C(C(x))^2 where C(x) = (1-sqrt(1-4*x))/2.

Examples

			G.f.: A(x) = x + x^2 + 3*x^3 + 14*x^4 + 84*x^5 + 592*x^6 + 4670*x^7 + 40108*x^8 + 368670*x^9 + 3586321*x^10 + ...
Related series.
A(x) = x + A(A(x))^2 - A(A(x))^3 where
A(A(x)) = x + 2*x^2 + 8*x^3 + 44*x^4 + 294*x^5 + 2244*x^6 + 18888*x^7 + ...
A(A(x))^2 = x^2 + 4*x^3 + 20*x^4 + 120*x^5 + 828*x^6 + 6368*x^7 + ...
A(A(x))^3 = x^3 + 6*x^4 + 36*x^5 + 236*x^6 + 1698*x^7 + ...

Links

Crossrefs

Cf. A368562, A213591.

Programs

  • PARI
    {a(n) = my(A=x+x^2); for(i=1,n, A = serreverse(x - A^2 + A^3 +x*O(x^n))); polcoeff(A,n)}
for(n=1,25,print1(a(n),", "))
  • PARI
    {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=x+x^2+x*O(x^n)); for(i=1, n, A= x + sum(m=1, n, Dx(m-1, A^(2*m)*(1 - A)^m )/m!) +x*O(x^n)); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Dec 30 2023

Formula

From Paul D. Hanna, Dec 30 2023: (Start)
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x) = x + A(A(x))^2 - A(A(x))^3.
(2) x = A( x - A(x)^2 + A(x)^3 ).
(3) x = A(A( x - x^2 + x^3 - A(x)^2 + A(x)^3 )).
(4) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(2*n) * (1 - A(x))^n / n!.
(5) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(2*n)*(1 - A(x))^n/x / n! ).
(6) A(x) is the unique solution to variable A in the infinite system of simultaneous equations starting with:
A = x + B^2 - B^3;
B = A + C^2 - C^3;
C = B + D^2 - D^3;
D = C + E^2 - E^3; ...
where B = A(A(x)), C = A(A(A(x))), D = A(A(A(A(x)))), etc.
... (End)
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