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A317354 G.f. satisfies: A(x) = 1 + x*A(x) * ( d/dx x*A(x)^4 ).

%I A317354 #7 Aug 09 2018 11:58:40
%S A317354 1,1,9,143,3077,81186,2500555,87388600,3402833181,145793616851,
%T A317354 6809846205490,344295007127226,18732966469820743,1091592281265899216,
%U A317354 67839235267146659208,4479984759698760684608,313349016085369145384269,23145049153829729002326633,1800552901345736894764054891,147168026141415485071183102785
%N A317354 G.f. satisfies: A(x) = 1 + x*A(x) * ( d/dx x*A(x)^4 ).
%H A317354 Vaclav Kotesovec, <a href="/A317354/b317354.txt">Table of n, a(n) for n = 0..362</a>
%F A317354 O.g.f. A(x) satisfies:
%F A317354 (1) A(x) = 1 + x*A(x) * ( d/dx x*A(x)^4 ).
%F A317354 (2) [x^n] exp( n * x*A(x)^4 ) / A(x) = 0 for n>0.
%F A317354 (3.a) [x^n] exp(-n * x*A(x)^4) * (2 - 1/A(x)) = 0 for n >= 1.
%F A317354 (3.b) [x^n] exp(-n^2 * x*A(x)^4) * (n + 1 - n/A(x)) = 0 for n >= 1.
%F A317354 (3.c) [x^n] exp(-n^(p+1) * x*A(x)^4) * (n^p + 1 - n^p/A(x)) = 0 for n>=1 and for fixed integer p >= 0.
%F A317354 a(n) ~ c * 4^n * n! * n^(5/4), where c = 0.1196141591155152938065262... - _Vaclav Kotesovec_, Aug 09 2018
%e A317354 O.g.f.: A(x) = 1 + x + 9*x^2 + 143*x^3 + 3077*x^4 + 81186*x^5 + 2500555*x^6 + 87388600*x^7 + 3402833181*x^8 + ...
%e A317354 where A(x) = 1 + x*A(x)^5 + 4*x^2*A(x)^4*A'(x).
%e A317354 RELATED TABLE.
%e A317354 The table of coefficients of x^k/k! in exp( n*x*A(x)^4 ) / A(x) begins:
%e A317354 n=1: [1, 0, -9, -554, -52995, -7440324, -1434977405, ...]
%e A317354 n=2: [1, 1, 0, -304, -38032, -5920008, -1205232128, ...];
%e A317354 n=3: [1, 2, 11, 0, -20475, -4189122, -949414689, ...];
%e A317354 n=4: [1, 3, 24, 364, 0, -2224296, -665082368, ...];
%e A317354 n=5: [1, 4, 39, 794, 23741, 0, -349575245, ...];
%e A317354 n=6: [1, 5, 56, 1296, 51120, 2511576, 0, ...];
%e A317354 n=7: [1, 6, 75, 1876, 82533, 5340642, 386787007, 0, ...]; ...
%e A317354 in which the coefficient of x^n in row n forms a diagonal of zeros.
%o A317354 (PARI) {a(n) = my(A=1); for(i=1,n, A = 1 + x*A*(x*A^4)' +x*O(x^n)); polcoeff(A,n)}
%o A317354 for(n=0,30, print1(a(n),", "))
%o A317354 (PARI) {a(n) = my(A=[1]); for(i=0,n, A=concat(A,0); m=#A-1; A[#A] = -Vec( exp(-m * x*Ser(A)^4 ) * (2 - 1/Ser(A)))[#A] ); A[n+1]}
%o A317354 for(n=0,30, print1(a(n),", "))
%o A317354 (PARI) {a(n) = my(A=[1]); for(i=0,n, A=concat(A,0); m=#A-1; A[#A] = -Vec( exp(-m^2 * x*Ser(A)^4 ) * (m+1 - m/Ser(A)))[#A]/m ); A[n+1]}
%o A317354 for(n=0,30, print1(a(n),", "))
%Y A317354 Cf.  A088716, A317352, A317353.
%K A317354 nonn
%O A317354 0,3
%A A317354 _Paul D. Hanna_, Jul 26 2018