A378257 G.f. A(x) satisfies A(x*A(x) - x^3*A'(x)) = x^2.

1, 1, 1, 3, 13, 65, 387, 2709, 21673, 195057, 1950547, 21456017, 257472123, 3347137599, 46859925639, 702898884585, 11246382147523, 191188496507891, 3441392937086571, 65386465804644849, 1307729316092328701, 27462315637938902721, 604170944034649445339, 13895931712796937242797

Offset

1,4

Comments

All terms are odd.

Limit a(n)/(n-1)! = 0.537518285445620493117044794174835470291338509726127442574604...

Links

Paul D. Hanna, Table of n, a(n) for n = 1..400

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

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

(2) A(x) = x^2*A'(x) + B(x^2)/x, where B(A(x)) = x.

(3) a(2*n) = (2*n-1)*a(2*n-1) for n >= 0.

(4) a(2*n+1) = (2*n)*a(2*n) + [x^(n+1)] B(x) for n >= 0, where B(A(x)) = x.

Example

G.f.: A(x) = x + x^2 + x^3 + 3*x^4 + 13*x^5 + 65*x^6 + 387*x^7 + 2709*x^8 + 21673*x^9 + 195057*x^10 + 1950547*x^11 + 21456017*x^12 + ...
where A(x*A(x) - x^3*A'(x)) = x^2.
RELATED SERIES.
The derivative of A(x) wrt x begins
A'(x) = 1 + 2*x + 3*x^2 + 12*x^3 + 65*x^4 + 390*x^5 + 2709*x^6 + 21672*x^7 + 195057*x^8 + 1950570*x^9 + 21456017*x^10 + ...
where [x^(2*n)] A'(x) = [x^(2*n+2)] A(x) for n >= 0.
Let B(x) satisfy B(A(x)) = x, then
B(x) = x - x^2 + x^3 - 3*x^4 + x^5 - 23*x^6 - 81*x^7 - 747*x^8 - 5837*x^9 - 55467*x^10 - 568279*x^11 - 6414523*x^12 + ...
where B(x^2) = x*A(x) - x^3*A'(x).

Prog

(PARI) {a(n) = my(V=[0, 1], A=x); for(i=1, n, V = concat(V, 0); A = Ser(V);
V[#V] = polcoef(x^2 - subst(A, x, x*A - x^3*A' ), #V) ); polcoef(A, n)}
for(n=1, 40, print1(a(n), ", "))

Crossrefs

Sequence in context: A232222 A241598 A155867 * A009102 A080227 A199143

Adjacent sequences: A378254 A378255 A378256 * A378258 A378259 A378260

Keyword

nonn,new

Author

Paul D. Hanna, Dec 04 2024

Status

approved