A295544 G.f. A(x) satisfies: x = 1 - A(x) - A(x)^2 + A(x)^4.

1, 1, -5, 46, -526, 6728, -92162, 1322296, -19616145, 298444410, -4631199738, 73016650304, -1166319788140, 18834433734788, -306979560915848, 5043390391728008, -83432990338737757, 1388625173325996954, -23235855361736469884, 390664036073212073184, -6596353435409461523030, 111808972896288124959480, -1901796869257241949592380, 32451148040799048565635240, -555335976299941700481867990, 9528782003452632755521672752

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..500

Formula

G.f. A(x) satisfies:

(1) A(x) = 1 + Series_Reversion( x + 5*x^2 + 4*x^3 + x^4 ).

(2) 1/A(x) = 1 - Series_Reversion( x*(1 + 2*x - 3*x^2 + x^3)/(1-x)^4 ).

(3) 1 + x = A( x + 5*x^2 + 4*x^3 + x^4 ).

(4) 1/(1-x) = A ( x*(1 + 2*x - 3*x^2 + x^3)/(1-x)^4 ).

(5) x = (1 - A(x)) * (1 - A(x)^2 - A(x)^3).

Example

G.f.: A(x) = 1 + x - 5*x^2 + 46*x^3 - 526*x^4 + 6728*x^5 - 92162*x^6 + 1322296*x^7 - 19616145*x^8 + 298444410*x^9 - 4631199738*x^10 + 73016650304*x^11 - 1166319788140*x^12 + 18834433734788*x^13 - 306979560915848*x^14 + 5043390391728008*x^15 - 83432990338737757*x^16 +...
such that x = 1 - A(x) - A(x)^2 + A(x)^4.
RELATED SERIES.
x/(1 - A(x)) = -1 - 5*x + 21*x^2 - 191*x^3 + 2177*x^4 - 27805*x^5 + 380565*x^6 - 5457258*x^7 + 80927961*x^8 - 1230920530*x^9 + 19097184623*x^10 +...
which equals 1 - A(x)^2 - A(x)^3.
1/A(x) = 1 - x + 6*x^2 - 57*x^3 + 659*x^4 - 8474*x^5 + 116437*x^6 - 1673929*x^7 + 24867461*x^8 - 378731849*x^9 + 5881775470*x^10 - 92792223330*x^11 +...
which equals 1 - Series_Reversion( x*(1 + 2*x - 3*x^2 + x^3)/(1-x)^4 ).
A(x)^2 = 1 + 2*x - 9*x^2 + 82*x^3 - 935*x^4 + 11944*x^5 - 163492*x^6 + 2344596*x^7 - 34770426*x^8 + 528876810*x^9 - 8205477566*x^10 +...
A(x)^3 = 1 + 3*x - 12*x^2 + 109*x^3 - 1242*x^4 + 15861*x^5 - 217073*x^6 + 3112662*x^7 - 46157535*x^8 + 702043720*x^9 - 10891707057*x^10 +...
A(x)^4 = 1 + 4*x - 14*x^2 + 128*x^3 - 1461*x^4 + 18672*x^5 - 255654*x^6 + 3666892*x^7 - 54386571*x^8 + 827321220*x^9 - 12836677304*x^10 +...
where A(x) = 1 - x - A(x)^2 + A(x)^4.

Prog

(PARI) {a(n) = my(A=1); A = 1 - serreverse( subst( 1 - x - x^2 + x^4, x, 1-x) +x^2*O(x^n) ); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); A = 1/(1 - serreverse( x*(2-x)/(1-x)^4 - x/(1-x) +x^2*O(x^n)) ); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A295543.

Sequence in context: A127304 A112029 A371380 * A058478 A367256 A159608

Adjacent sequences: A295541 A295542 A295543 * A295545 A295546 A295547

Keyword

sign

Author

Paul D. Hanna, Nov 24 2017

Status

approved