This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A351920 #5 Feb 28 2022 08:09:53
%S A351920 1,1,-2,15,-200,3920,-102924,3424946,-139217280,6733296720,
%T A351920 -379945682400,24634909864752,-1813131434947392,149981854962931680,
%U A351920 -13828816882622028000,1411324560147609680400
%N A351920 E.g.f. A(x) satisfies: A(x/A(x)) = A(x) / (A(x) - x*A(x)').
%C A351920 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.
%F A351920 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies:
%F A351920 (1) A(x/A(x)) = A(x) / (A(x) - x*A(x)').
%F A351920 (2) a(n) = [x^n/n!] A(x) = [x^n/n!] A(x)^n, for n >= 0.
%F A351920 (3) A(x) = x*A'(x) / (1 - 1/A(x/A(x))).
%e A351920 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! + ...
%e A351920 where A(x/A(x)) = A(x) / (A(x) - x*A(x)').
%e A351920 Related table.
%e A351920 Here we illustrate the related formula
%e A351920 a(n) = [x^n/n!] A(x) = [x^n/n!] A(x)^n (n >= 0).
%e A351920 The table of coefficients of x^k/k! in A(x)^n begins:
%e A351920 n=0: [1, 0, 0, 0, 0, 0, 0, 0, ...];
%e A351920 n=1: [1, 1, -2, 15, -200, 3920, -102924, 3424946, ...];
%e A351920 n=2: [1, 2, -2, 18, -256, 5240, -142308, 4869676, ...];
%e A351920 n=3: [1, 3, 0, 15, -240, 5220, -147672, 5212410, ...];
%e A351920 n=4: [1, 4, 4, 12, -200, 4640, -137016, 4992008, ...];
%e A351920 n=5: [1, 5, 10, 15, -160, 3920, -120420, 4521370, ...];
%e A351920 n=6: [1, 6, 18, 30, -120, 3240, -102924, 3970596, ...];
%e A351920 n=7: [1, 7, 28, 63, -56, 2660, -86688, 3424946, ...]; ...
%e A351920 in which the main diagonal equals this sequence and is found in row n = 1.
%e A351920 Related series.
%e A351920 Notice the relation to A179421, given by
%e A351920 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! + ...
%e A351920 Also, the following composition of functions
%e A351920 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! + ...
%e A351920 equals A(x) / (A(x) - x*A(x)'), as specified in the definition.
%o A351920 (PARI) {a(n) = my(A=[1,1]); for(m=1,n, A=concat(A,0);
%o A351920 A[#A] = -Vec(Ser(A)^(#A-1))[#A]/(#A-2) ); n!*A[n+1]}
%o A351920 for(n=0,12,print1(a(n),", "))
%Y A351920 Cf. A179421, A179420.
%K A351920 sign
%O A351920 0,3
%A A351920 _Paul D. Hanna_, Feb 25 2022