A213192 G.f. satisfies: A(x) = 1 + x * sqrt( d/dx x*A(x)^4 ).

1, 1, 4, 25, 204, 2024, 23560, 315147, 4777932, 81249562, 1536125144, 32033803936, 731461308600, 18165906647652, 487702581895824, 14076885317410829, 434705720055275532, 14300349927653656920, 499229708336982490936, 18432449956359308918034, 717597821570439742670440

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

G.f. satisfies: A(x) = 1 + x * sqrt( A(x)^4 + 4*x*A(x)^3*A'(x) ).

a(m) == 1 (mod 2) at m = 2^n-1 for n>=0, otherwise a(m) == 0 (mod 2).

a(n) ~ c * 2^n * n! / sqrt(n), where c = 1.2412292448741911566... - Vaclav Kotesovec, Aug 24 2017

Example

G.f.: A(x) = 1 + x + 4*x^2 + 25*x^3 + 204*x^4 + 2024*x^5 + 23560*x^6 +...
Related expansions:
A(x)^4 = 1 + 4*x + 22*x^2 + 152*x^3 + 1261*x^4 + 12252*x^5 + 137370*x^6 +...
d/dx x*A(x)^4 = 1 + 8*x + 66*x^2 + 608*x^3 + 6305*x^4 + 73512*x^5 +...
(A(x)-1)^2 = x^2 + 8*x^3 + 66*x^4 + 608*x^5 + 6305*x^6 + 73512*x^7 +...

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x; for(i=1, n, A=1+x*deriv(x*(A+x*O(x^n))^4)^(1/2))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A226067.

Sequence in context: A347585 A088159 A301363 * A036242 A120955 A061714

Adjacent sequences: A213189 A213190 A213191 * A213193 A213194 A213195

Keyword

nonn

Author

Paul D. Hanna, Feb 28 2013

Status

approved