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 A351918 #10 Feb 28 2022 13:38:13
%S A351918 1,2,16,262,6688,231938,10130888,532099310,32582983264,2275724862530,
%T A351918 178357712565096,15489332978290006,1475661504819976992,
%U A351918 152969444613431797250,17137064243138924753480,2062951616302994096028894
%N A351918 G.f. A(x) satisfies: [x^n] (1 + n*x - A(x))^(2*n+1) = 0, for n > 0.
%C A351918 Compare to: [x^n] (1 + n*x - C(x))^(n+1) = 0, for n>0, where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers
%C A351918 Compare to: [x^n] (1 + n*x - W(x))^n = 0, for n>0, where W(x) = Sum_{n>=1} (n-1)^(n-1)*x^n/n! = 1 + x/LambertW(-x).
%H A351918 Vaclav Kotesovec, <a href="/A351918/b351918.txt">Table of n, a(n) for n = 1..330</a>
%F A351918 a(n) ~ c * d^n * n! / n^2, where d = (1+r) / ((-1 + exp(r + LambertW(-1, -exp(-r)*r))) * LambertW(-exp(-1-r)*(1+r))) = 8.406107401279769476199925123910168..., r = 0.7545302104650497245839827141610818561001159135034... is the root of the equation r*(1 + r + LambertW(-exp(-1 - r)*(1 + r))) = -(1 + r)*(r + LambertW(-1, -exp(-r)*r)) and c = 0.053954066... - _Vaclav Kotesovec_, Feb 27 2022
%e A351918 G.f.: A(x) = x + 2*x^2 + 16*x^3 + 262*x^4 + 6688*x^5 + 231938*x^6 + 10130888*x^7 + 532099310*x^8 + 32582983264*x^9 + 2275724862530*x^10 + ...
%e A351918 Related table.
%e A351918 Here we illustrate the formula [x^n] (1 + n*x - A(x))^(2*n+1) = 0 (n > 0).
%e A351918 The table of coefficients of x^k in (1 + n*x - A(x))^(2*n+1) begins:
%e A351918 n=0: [1, -1, -2, -16, -262, -6688, -231938, ...];
%e A351918 n=1: [1, 0, -6, -48, -774, -19872, -691910, ...];
%e A351918 n=2: [1, 5, 0, -110, -1645, -38439, -1286640, ...];
%e A351918 n=3: [1, 14, 70, 0, -3374, -74928, -2272326, ...];
%e A351918 n=4: [1, 27, 306, 1692, 0, -144342, -4262946, ...];
%e A351918 n=5: [1, 44, 858, 9504, 58938, 0, -7887286, ...];
%e A351918 n=6: [1, 65, 1924, 33982, 388401, 2694523, 0, ...]; ...
%e A351918 in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1 + n*x - A(x))^(2*n+1) = 0, for n > 0.
%o A351918 (PARI) {a(n) = my(A=[1], m=1); for(i=1, n, A=concat(A, 0); m=#A;
%o A351918 A[m] = polcoeff( (1 + m*x - x*Ser(A))^(2*m+1), m)/(2*m+1) ); A[n]}
%o A351918 for(n=1, 20, print1(a(n), ", "))
%Y A351918 Cf. A338408, A337758.
%K A351918 nonn
%O A351918 1,2
%A A351918 _Paul D. Hanna_, Feb 25 2022