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 A300988 #24 Mar 19 2018 21:56:16
%S A300988 1,1,3,43,1369,69561,4991371,471516403,56029153713,8112993527089,
%T A300988 1398528216254611,281935928284459131,65543089930613822473,
%U A300988 17373185629100099938153,5201713100466658289659419,1745470558150260528082445251,652016607740826946854349450081,269558306371535265856134699842913,122707064351998882900943162086492963,61225312946191234549695844364141862859
%N A300988 E.g.f. A(x) satisfies: [x^n] A(x)^(4*n) = (n + 3) * [x^(n-1)] A(x)^(4*n) for n>=1.
%C A300988 Compare to: [x^n] exp(x)^(4*n) = 4 * [x^(n-1)] exp(x)^(4*n) for n>=1.
%H A300988 Paul D. Hanna, <a href="/A300988/b300988.txt">Table of n, a(n) for n = 0..200</a>
%F A300988 E.g.f. A(x) satisfies: A(x) = exp( x * (A(x) - 3*x*A'(x)) / (A(x) - 4*x*A'(x)) ).
%e A300988 E.g.f.: A(x) = 1 + x + 3*x^2/2! + 43*x^3/3! + 1369*x^4/4! + 69561*x^5/5! + 4991371*x^6/6! + 471516403*x^7/7! + 56029153713*x^8/8! + 8112993527089*x^9/9! + ...
%e A300988 such that [x^n] A(x)^(4*n) = (n+3) * [x^(n-1)] A(x)^(4*n) for n>=1.
%e A300988 RELATED SERIES.
%e A300988 A(x)^4 = 1 + 4*x + 24*x^2/2! + 304*x^3/3! + 8320*x^4/4! + 390144*x^5/5! + 26653696*x^6/6! + 2434011136*x^7/7! + 282056564736*x^8/8! + ...
%e A300988 ILLUSTRATION OF DEFINITION.
%e A300988 The table of coefficients of x^k in A(x)^(4*n) begins:
%e A300988 n=1: [(1), (4), 12, 152/3, 1040/3, 16256/5, 1665856/45, 152125696/315, ...];
%e A300988 n=2: [1, (8), (40), 592/3, 3728/3, 157376/15, 4992064/45, 86636800/63, ...];
%e A300988 n=3: [1, 12, (84), (504), 3264, 129408/5, 1273536/5, 104486784/35, ...];
%e A300988 n=4: [1, 16, 144, (3104/3), (21728/3), 283264/5, 23764096/45, 1844359168/315, ...];
%e A300988 n=5: [1, 20, 220, 5560/3, (42800/3), (342400/3), 9296960/9, 687731200/63, ...];
%e A300988 n=6: [1, 24, 312, 3024, 25680, (1073856/5), (9664704/5), 690265344/35, ...];
%e A300988 n=7: [1, 28, 420, 13832/3, 129248/3, 1905792/5, (156447424/45), (312894848/9), ...]; ...
%e A300988 in which the coefficients in parenthesis are related by
%e A300988 4 = 4*(1); 40 = 5*(8); 504 = 6*(84); 21728/3 = 7*(3104/3); 342400/3 = 8*(42800/3); 9664704/5 = 9*(1073856/5); ...
%e A300988 illustrating: [x^n] A(x)^(4*n) = (n+3) * [x^(n-1)] A(x)^(4*n).
%e A300988 LOGARITHMIC PROPERTY.
%e A300988 The logarithm of the e.g.f. is an integer power series in x satisfying
%e A300988 log(A(x)) = x * (1 - 3*x*A'(x)/A(x)) / (1 - 4*x*A'(x)/A(x));
%e A300988 explicitly,
%e A300988 log(A(x)) = x + x^2 + 6*x^3 + 50*x^4 + 520*x^5 + 6312*x^6 + 86080*x^7 + 1288704*x^8 + 20862720*x^9 + 361454720*x^10 + ... + A300989(n)*x^n + ...
%o A300988 (PARI) {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(4*(#A-1))); A[#A] = ((#A+2)*V[#A-1] - V[#A])/(4*(#A-1)) ); n!*polcoeff( Ser(A), n)}
%o A300988 for(n=0, 25, print1(a(n), ", "))
%o A300988 (PARI) {a(n) = my(A=1); for(i=1,n, A = exp( x*(A-3*x*A')/(A-4*x*A' +x*O(x^n)) ) ); n!*polcoeff(A,n)}
%o A300988 for(n=0, 25, print1(a(n), ", "))
%Y A300988 Cf. A300989, A182962, A300735, A300986, A300990, A300992.
%Y A300988 Cf. A300735, A300870, A300590, A296170.
%K A300988 nonn
%O A300988 0,3
%A A300988 _Paul D. Hanna_, Mar 17 2018