A324299 G.f. A(x) satisfies: 1 + 2*x = Sum_{n>=0} (2^n + q*A(x))^n * x^n / (1 + 2^n*q*x*A(x))^(n+1), where q = sqrt(8).

1, 4, 1244, 855536, 1998222360, 17176861671264, 572050564720167008, 75376047180612855646592, 39584783503733394267753112672, 83058214142365093436392867414792576, 696852099724215401960179406060140816970880, 23383571828415218844195068102736723566962055422464, 3138533189471836170041724947584184029965599257158176401152, 1684993942439400740133233780973963561210723646789741914125107608576, 3618501123890248279401995297309664867635750984620022599181205794552819346432, 31082698237363127668144117571836989206875241298135354103905143740745373368280843792384

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..50

Formula

Let q = sqrt(8), then g.f. A(x) satisfies:

(1) 1 + 2*x = Sum_{n>=0} (2^n + q * A(x))^n * x^n / (1 + 2^n * q * x*A(x))^(n+1).

(2) 1 + 2*x = Sum_{n>=0} (2^n - q * A(x))^n * x^n / (1 - 2^n * q * x*A(x))^(n+1).

Example

G.f.: A(x) = 1 + 4*x + 1244*x^2 + 855536*x^3 + 1998222360*x^4 + 17176861671264*x^5 + 572050564720167008*x^6 + 75376047180612855646592*x^7 + ...
Let q = sqrt(8), then
1 + 2*x = 1/(1 + x*q*A(x)) + (2 + q*A(x))*x/(1 + 2*x*q*A(x))^2 + (2^2 + q*A(x))^2*x^2/(1 + 2^2*x*q*A(x))^3 + (2^3 + q*A(x))^3*x^3/(1 + 2^3*x*q*A(x))^4 + (2^4 + q*A(x))^4*x^4/(1 + 2^4*x*q*A(x))^5 + (2^5 + q*A(x))^5*x^5/(1 + 2^5*x*q*A(x))^6 + (2^6 + q*A(x))^6*x^6/(1 + 2^6*x*q*A(x))^7 + ...
and also
1 + 2*x = 1/(1 - x*q*A(x)) + (2 - q*A(x))*x/(1 - 2*x*q*A(x))^2 + (2^2 - q*A(x))^2*x^2/(1 - 2^2*x*q*A(x))^3 + (2^3 - q*A(x))^3*x^3/(1 - 2^3*x*q*A(x))^4 + (2^4 - q*A(x))^4*x^4/(1 - 2^4*x*q*A(x))^5 + (2^5 - q*A(x))^5*x^5/(1 - 2^5*x*q*A(x))^6 + (2^6 - q*A(x))^6*x^6/(1 - 2^6*x*q*A(x))^7 + ...

Prog

(PARI) /* Requires high precision */
{a(n) = my(q=sqrt(8), A=[1, 4, 0]); for(i=0, n,
A=concat(A, 0); A[#A-1] = round( polcoeff( sum(n=0, #A, (2^n + q * Ser(A))^n * x^n / (1 + 2^n * q * x*Ser(A))^(n+1) ), #A)/32)); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A324613.

Sequence in context: A350341 A253168 A160004 * A227850 A079711 A283260

Adjacent sequences: A324296 A324297 A324298 * A324300 A324301 A324302

Keyword

nonn

Author

Paul D. Hanna, Mar 10 2019

Status

approved