A351920 E.g.f. A(x) satisfies: A(x/A(x)) = A(x) / (A(x) - x*A(x)').

1, 1, -2, 15, -200, 3920, -102924, 3424946, -139217280, 6733296720, -379945682400, 24634909864752, -1813131434947392, 149981854962931680, -13828816882622028000, 1411324560147609680400

Offset

0,3

Comments

Given e.g.f. A(x) of this sequence, 1/A(x) equals the e.g.f. of A179421 - see entry A179421 for further formulas.

Links

Table of n, a(n) for n=0..15.

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies:

(1) A(x/A(x)) = A(x) / (A(x) - x*A(x)').

(2) a(n) = [x^n/n!] A(x) = [x^n/n!] A(x)^n, for n >= 0.

(3) A(x) = x*A'(x) / (1 - 1/A(x/A(x))).

Example

E.g.f.: A(x) = 1 + x - 2*x^2/2! + 15*x^3/3! - 200*x^4/4! + 3920*x^5/5! - 102924*x^6/6! + 3424946*x^7/7! - 139217280*x^8/8! + ...
where A(x/A(x)) = A(x) / (A(x) - x*A(x)').
Related table.
Here we illustrate the related formula
a(n) = [x^n/n!] A(x) = [x^n/n!] A(x)^n (n >= 0).
The table of coefficients of x^k/k! in A(x)^n begins:
n=0: [1, 0,  0,   0,    0,    0,       0,       0, ...];
n=1: [1, 1, -2,  15, -200, 3920, -102924, 3424946, ...];
n=2: [1, 2, -2,  18, -256, 5240, -142308, 4869676, ...];
n=3: [1, 3,  0,  15, -240, 5220, -147672, 5212410, ...];
n=4: [1, 4,  4,  12, -200, 4640, -137016, 4992008, ...];
n=5: [1, 5, 10,  15, -160, 3920, -120420, 4521370, ...];
n=6: [1, 6, 18,  30, -120, 3240, -102924, 3970596, ...];
n=7: [1, 7, 28,  63,  -56, 2660,  -86688, 3424946, ...]; ...
in which the main diagonal equals this sequence and is found in row n = 1.
Related series.
Notice the relation to A179421, given by
1/A(x) = 1 - x + 4*x^2/2! - 33*x^3/3! + 440*x^4/4! - 8380*x^5/5! + 211824*x^6/6! - 6771422*x^7/7! + ... + A179421(n)*x^n/n! + ...
Also, the following composition of functions
A(x/A(x)) = 1 + x - 4*x^2/2! + 39*x^3/3! - 632*x^4/4! + 14620*x^5/5! - 445104*x^6/6! + 16958522*x^7/7! - 781426848*x^8/8! + ...
equals A(x) / (A(x) - x*A(x)'), as specified in the definition.

Prog

(PARI) {a(n) = my(A=[1, 1]); for(m=1, n, A=concat(A, 0);
A[#A] = -Vec(Ser(A)^(#A-1))[#A]/(#A-2) ); n!*A[n+1]}
for(n=0, 12, print1(a(n), ", "))

Crossrefs

Cf. A179421, A179420.

Sequence in context: A042355 A208467 A221102 * A319466 A020557 A323118

Adjacent sequences: A351917 A351918 A351919 * A351921 A351922 A351923

Keyword

sign

Author

Paul D. Hanna, Feb 25 2022

Status

approved