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A375451 Expansion of g.f. A(x) satisfying 0 = Sum_{k=0..n} (-1)^k * binomial(n,k)^2 * ([x^k] A(x)^n) for n >= 1.

%I A375451 #8 Sep 10 2024 12:57:19
%S A375451 1,1,3,21,264,5100,138595,5021209,233863116,13628372628,972514037307,
%T A375451 83479400425677,8490972592164813,1010263560882000981,
%U A375451 139051185192340895094,21926159523172792097194,3927328317712845680689864,793059545751159815604109176,179339266160209677707004583560
%N A375451 Expansion of g.f. A(x) satisfying 0 = Sum_{k=0..n} (-1)^k * binomial(n,k)^2 * ([x^k] A(x)^n) for n >= 1.
%C A375451 Note that 0 = Sum_{k=0..n} (-1)^k * binomial(n,k) * ([x^k] G(x)^n) is satisfied by G(x) = 1/(1-x) for n >= 1.
%H A375451 Paul D. Hanna, <a href="/A375451/b375451.txt">Table of n, a(n) for n = 0..200</a>
%F A375451 a(n) ~ c * d^n * n!^2, where d = 0.691660276122579707675... = 4/BesselJZero(0,1)^2 = 4/A115368^2 and c = 3.8999463598998648630203... - _Vaclav Kotesovec_, Sep 10 2024
%e A375451 G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 264*x^4 + 5100*x^5 + 138595*x^6 + 5021209*x^7 + 233863116*x^8 + ...
%e A375451 The table of coefficients of x^k in A(x)^n begins:
%e A375451   n=1: [1, 1,  3,  21,  264,  5100,  138595, ...];
%e A375451   n=2: [1, 2,  7,  48,  579, 10854,  289415, ...];
%e A375451   n=3: [1, 3, 12,  82,  954, 17352,  453657, ...];
%e A375451   n=4: [1, 4, 18, 124, 1399, 24696,  632656, ...];
%e A375451   n=5: [1, 5, 25, 175, 1925, 33001,  827900, ...];
%e A375451   n=6: [1, 6, 33, 236, 2544, 42396, 1041046, ...];
%e A375451   ...
%e A375451 from which we may illustrate the defining property given by
%e A375451 0 = Sum_{k=0..n} (-1)^k * binomial(n,k)^2 * ([x^k] A(x)^n).
%e A375451 Using the coefficients in the table above, we see that
%e A375451   n=1: 0 = 1*1 - 1*1;
%e A375451   n=2: 0 = 1*1 - 4*2 + 1*7;
%e A375451   n=3: 0 = 1*1 - 9*3 + 9*12 - 1*82;
%e A375451   n=4: 0 = 1*1 - 16*4 + 36*18 - 16*124 + 1*1399;
%e A375451   n=5: 0 = 1*1 - 25*5 + 100*25 - 100*175 + 25*1925 - 1*33001;
%e A375451   n=6: 0 = 1*1 - 36*6 + 225*33 - 400*236 + 225*2544 - 36*42396 + 1*1041046;
%e A375451   ...
%o A375451 (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A375451 A[#A] = sum(k=0,#A-1, (-1)^(#A-k) * binomial(#A-1,k)^2 * polcoef(Ser(A)^(#A-1),k) )/(#A-1) ); A[n+1]}
%o A375451 for(n=0,20,print1(a(n),", "))
%Y A375451 Cf. A008459, A115368.
%K A375451 nonn
%O A375451 0,3
%A A375451 _Paul D. Hanna_, Sep 10 2024