A236956 E.g.f. satisfies: A'(x) = A(x)^7 * A(-x)^3 with A(0) = 1.

1, 1, 4, 46, 568, 11920, 249760, 7572880, 224573440, 8889771520, 341458163200, 16665769120000, 786624917478400, 45625565188403200, 2555686067488153600, 171677981964256768000, 11132816984129032192000, 849791969438296428544000, 62624820152427866324992000, 5352961417325349446434816000

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..359

Formula

E.g.f.: 1/(1 - 3*x*G(3*x^2))^(1/3) where G(x) = 1 + x*G(x)^3 = g.f. of A001764.

E.g.f.: 1/(1 - 3*Series_Reversion( Integral (1 - 9*x^2) dx ))^(1/3).

From Robert Israel, Jun 08 2026: (Start)

E.g.f. A(x) satisfies 9*x*A(x)^9 - 2*A(x)^9 + 3*A(x)^3 - 1 = 0.

D-finite with recurrence (729*n^3 + 972*n^2 + 351*n + 28)*a(n) + (-162*n^2 - 324*n - 204)*a(n + 1) + (-36*n - 48)*a(n + 2) + 8*a(n + 3) = 0. (End)

Example

E.g.f.: A(x) = 1 + x + 4*x^2/2! + 46*x^3/3! + 568*x^4/4! + 11920*x^5/5! +...
Related series.
A(x)^7 = 1 + 7*x + 77*x^2/2! + 1267*x^3/3! + 26873*x^4/4! + 712831*x^5/5! +...
A(x)^3 = 1 + 3*x + 9*x^2 + 36*x^3 + 135*x^4 + 567*x^5 + 2268*x^6 + 9720*x^7 +... where A(x)^3 = 1/(1 - 3*x*G(3*x^2)) where G(x) = 1 + x*G(x)^3.
Note that 1 - 1/A(x)^3 is an odd function:
1 - 1/A(x)^3 = 3*x + 9*x^3 + 81*x^5 + 972*x^7 + 13365*x^9 + 199017*x^11 +...
where Series_Reversion((1 - 1/A(x)^3)/3) = Integral (1-9*x^2) dx.

Maple

f:= gfun:-rectoproc({(729*n^3 + 972*n^2 + 351*n + 28)*a(n) + (-162*n^2 - 324*n - 204)*a(n + 1) + (-36*n - 48)*a(n + 2) + 8*a(n + 3), a(0)=1, a(1)=1, a(2)=4}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Jun 08 2026

Prog

(PARI) {a(n)=local(A=1); for(i=0, n, A=1+intformal(A^7*subst(A^3, x, -x) +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(A=1); A=1/(1-3*serreverse(intformal((1-9*x^2 +x*O(x^n))^(3/3))))^(1/3); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A236953, A236954, A236955, A235373, A236957, A236958.

Sequence in context: A197131 A235132 A385473 * A394709 A113264 A264717

Adjacent sequences: A236953 A236954 A236955 * A236957 A236958 A236959

Keyword

nonn

Author

Paul D. Hanna, Apr 09 2014

Status

approved