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

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

1,4

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

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).

Example

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).

Prog

(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), ", "))

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.

Cf. A277300, A277301, A277302, A277303, A277304, A277305, A277307, A277308, A277309, A277310, A277311.

Cf. A276364.

Sequence in context: A107667 A362155 A163176 * A201444 A236381 A335300

Adjacent sequences: A277303 A277304 A277305 * A277307 A277308 A277309

Keyword

sign

Author

Paul D. Hanna, Oct 09 2016

Status

approved