A193342 E.g.f.: A(x) = G(x)*exp(-x/2)/x where G(x) satisfies: G(G(x)) = x*exp(G(x)), and A(x) = Sum_{n>=0} a(n)*x^(2*n)/((2*n)!*4^n).

1, 1, -7, 873, -335023, 314308145, -608475110391, 2176841249613401, -13293673514920102879, 130392618478782066711009, -1956708639203083689685074535, 43167469497976800185127921454793, -1354293569879914292359532215444184463, 58748391997267678043451322126451570916113

Offset

0,3

Comments

It is surprising that the e.g.f. of this sequence is an even function.

Links

Table of n, a(n) for n=0..13.

Example

G.f.: A(x) = 1 + 1*x^2/(2!*2^2) - 7*x^4/(4!*2^4) + 873*x^6/(6!*2^6) - 335023*x^8/(8!*2^8) + 314308145*x^10/(10!*2^10) - 608475110391*x^12/(12!*2^12) + 2176841249613401*x^14/(14!*2^14) +...
where G(x) = x*A(x)*exp(x/2) satisfies G(G(x)) = x*exp(G(x)):
G(x) = x + 2*x^2/(2!*2) + 6*x^3/(3!*4) + 16*x^4/(4!*8) - 144*x^6/(6!*32) + 5488*x^7/(7!*64) + 47104*x^8/(8!*128) - 2799360*x^9/(9!*256) - 29427200*x^10/(10!*512) +...
and is the e.g.f. of A193341.

Prog

(PARI) {a(n)=local(A=x+x^2); for(i=1, 2*n, A=A+(x*exp(A+O(x^(2*n+1)))-subst(A, x, A))/2); if(n<0, 0, (2*n)!*4^n*polcoeff(A/x*exp(-x/2+O(x^(2*n+1))), 2*n))}

Crossrefs

Cf. A193341.

Sequence in context: A269896 A087350 A308296 * A298301 A332187 A093171

Adjacent sequences: A193339 A193340 A193341 * A193343 A193344 A193345

Keyword

sign

Author

Paul D. Hanna, Jul 23 2011

Status

approved