A273925 -id:A273925 - cp's OEIS frontend

A273926 Given G(x) such that G( G(x)^2 - G(x)^3 ) = x^2, then G(x) = Sum_{n>=1} A273925(n)*x^n / 2^a(n).

0, 1, 3, 2, 7, 4, 10, 5, 15, 8, 18, 9, 22, 11, 25, 12, 31, 16, 34, 17, 38, 19, 41, 20, 46, 23, 49, 24, 53, 26, 56, 27, 63, 32, 66, 33, 70, 35, 73, 36, 78, 39, 81, 40, 85, 42, 88, 43, 94, 47, 97, 48, 101, 50, 104, 51, 109, 54, 112, 55, 116, 57, 119, 58, 127, 64, 130, 65, 134, 67, 137, 68, 142, 71, 145, 72, 149, 74, 152, 75, 158, 79, 161, 80, 165, 82, 168, 83, 173, 86, 176, 87, 180, 89, 183, 90, 190, 95, 193, 96, 197, 98, 200, 99, 205, 102, 208, 103, 212, 105, 215, 106, 221, 110, 224, 111, 228, 113, 231, 114

Offset: 1

Paul D. Hanna, Jun 04 2016

nonn

Terms appear to occur only once in the sequence.

Both bisections of this sequence appear to be monotonically increasing.

			G.f.: G(x) = x + 1/2*x^2 + 3/8*x^3 + 3/4*x^4 + 175/128*x^5 + 41/16*x^6 + 4947/1024*x^7 + 321/32*x^8 + 687611/32768*x^9 + 11403/256*x^10 + 25132181/262144*x^11 + 107305/512*x^12 + 1941554203/4194304*x^13 + 2111325/2048*x^14 + 77643067507/33554432*x^15 + 21427329/4096*x^16 + 25549683166419/2147483648*x^17 + 1782548851/65536*x^18 + 1073363084982753/17179869184*x^19 + 18891311061/131072*x^20 + 91744420207896017/274877906944*x^21 + 406630578535/524288*x^22 + 3975787925128277349/2199023255552*x^23 + 4432136534071/1048576*x^24 +...+ A273925(n)*x^n / 2^a(n) +...
such that G( G(x)^2 - G(x)^3 ) = x^2.
The bisections of this sequence begin:
odd bisection (cf. A120738): [0, 3, 7, 10, 15, 18, 22, 25, 31, 34, 38, 41, 46, 49, 53, 56, 63, 66, 70, 73, 78, 81, 85, 88, 94, 97, 101, 104, 109, 112, 116, 119, 127, 130, 134, 137, 142, 145, 149, 152, 158, 161, 165, 168, 173, 176, 180, 183, 190, 193, 197, 200, 205, 208, 212, 215, 221, 224, 228, 231, 236, 239, 243, 246, 255, ...].
even bisection (cf. A101925): [1, 2, 4, 5, 8, 9, 11, 12, 16, 17, 19, 20, 23, 24, 26, 27, 32, 33, 35, 36, 39, 40, 42, 43, 47, 48, 50, 51, 54, 55, 57, 58, 64, 65, 67, 68, 71, 72, 74, 75, 79, 80, 82, 83, 86, 87, 89, 90, 95, 96, 98, 99, 102, 103, 105, 106, 110, 111, 113, 114, 117, 118, 120, 121, 128, ...].

Paul D. Hanna, Table of n, a(n) for n = 1..261

Cf. A273925, A120738, A101925, A000120, A005187.

{a(n) = my(A=x); for(i=0,n, A = serreverse( sqrt(subst(A,x,x^2 - x^3 +x^2*O(x^n) )) )); valuation(denominator(polcoeff(A,n)),2)}
for(n=1,60,print1(a(n),", "))

a(2*n-1) = A120738(n-1) = 4*(n-1) - A000120(n-1), for n>=0 (conjecture).

a(2*n) = A101925(n-1) = A005187(n-1) + 1, for n>=0 (conjecture).

A367384 Expansion of g.f. A(x) satisfying A( sqrt(A(x)^2 - 8*A(x)^3) ) = x.

Original entry on oeis.org

1, 2, 16, 172, 2120, 28264, 396192, 5746480, 85394656, 1291778368, 19805198784, 306834276416, 4793670528640, 75415927948416, 1193652980090880, 18994846756882176, 303766882134726144, 4880209392051146752, 78739290124904116224, 1275444751485628848128, 20735204112205333970944

Offset: 1

Paul D. Hanna, Dec 29 2023

nonn

			G.f.: A(x) = x + 2*x^2 + 16*x^3 + 172*x^4 + 2120*x^5 + 28264*x^6 + 396192*x^7 + 5746480*x^8 + 85394656*x^9 + 1291778368*x^10 + ...
where A( sqrt(A(x)^2 - 8*A(x)^3) ) = x.
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 36*x^4 + 408*x^5 + 5184*x^6 + 70512*x^7 + 1002864*x^8 + 14711456*x^9 + 220670592*x^10 + ...
A(x)^3 = x^3 + 6*x^4 + 60*x^5 + 716*x^6 + 9384*x^7 + 130344*x^8 + 1882576*x^9 + 27950736*x^10 + ...
Let Ai(x) be the series reversion of A(x), then
Ai(x)^2 = A(x)^2 - 8*A(x)^3 = x^2 - 4*x^3 - 12*x^4 - 72*x^5 - 544*x^6 - 4560*x^7 - 39888*x^8 - 349152*x^9 - 2935296*x^10 - ...
and
Ai(x) = sqrt(A(x)^2 - 8*A(x)^3) = x - 2*x^2 - 8*x^3 - 52*x^4 - 408*x^5 - 3512*x^6 - 31584*x^7 - 287056*x^8 - 2560288*x^9 - ...
Also,
A(A(x)) = x + 4*x^2 + 40*x^3 + 512*x^4 + 7392*x^5 + 114688*x^6 + 1867008*x^7 + 31457280*x^8 + 543921664*x^9 + ... + 2^n*A078531(n)*x^(n+1) + ...
which satisfies A(A(x))^2 - 8*A(A(x))^3 = x^2, where
A(A(x))^2 = x^2 + 8*x^3 + 96*x^4 + 1344*x^5 + 20480*x^6 + 329472*x^7 + ...
A(A(x))^3 = x^3 + 12*x^4 + 168*x^5 + 2560*x^6 + 41184*x^7 + 688128*x^8 + ...

Cf. A078531, A273925.

{a(n) = my(A=1,V=[1]); for(i=1,n, V = concat(V,0); A = x*Ser(V);
V[#V] = polcoeff( x - subst(A,x, sqrt(A^2 - 8*A^3)), #V)/2 );V[n]}
for(n=1,30,print1(a(n),", "))

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) x = A( sqrt(A(x)^2 - 8*A(x)^3) ).
(2) x^2 = A(A(x))^2 - 8*A(A(x))^3, where 2*A(A(x/2)) is the g.f. of A078531.
(3) [x^(n+1)] A(A(x)) = 8^n * binomial((3*n-1)/2, n)/(n+1) = 2^n*A078531(n) for n >= 0.

cp's OEIS Frontend

A273926 Given G(x) such that G( G(x)^2 - G(x)^3 ) = x^2, then G(x) = Sum_{n>=1} A273925(n)*x^n / 2^a(n).

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