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A302107 G.f. A(x) satisfies: [x^(n+1)] A(x)^(n^2) / (x*A(x)^n)' = 0 for n>=0. 0

%I

%S 1,1,1,1,7,211,8411,412301,24894581,1832290133,162840289853,

%T 17318483860823,2184687906940713,323935018037153709,

%U 55939700211256251813,11149440249347239643775,2542901827027280314993359,658411118385997264277976111,192135231563520713206597464479,62778202074038700301319695876973,22831139565407893815484402030333403

%N G.f. A(x) satisfies: [x^(n+1)] A(x)^(n^2) / (x*A(x)^n)' = 0 for n>=0.

%C Compare to: [x^(n-1)] (x*F(x)^n)' / F(x)^(n^2) = 0 for n>0 holds when F(0) = 1.

%F G.f. A(x) satisfies: [x^(n+1)] A(x)^(n^2-n+1) / (A(x) + n*x*A'(x)) = 0 for n>=0.

%e G.f.: A(x) = 1 + x + x^2 + x^3 + 7*x^4 + 211*x^5 + 8411*x^6 + 412301*x^7 + 24894581*x^8 + 1832290133*x^9 + 162840289853*x^10 + ...

%e ILLUSTRATION OF DEFINITION.

%e The table of coefficients of x^k in A(x)^(n^2) / (x*A(x)^n)' begins:

%e n=0: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...];

%e n=1: [1, -1, 0, 0, -24, -972, -47184, -2729850, -190252260, ...];

%e n=2: [1, 0, 1, 0, -35, -1536, -78051, -4655400, -331711815, ...];

%e n=3: [1, 3, 9, 17, 0, -1674, -94734, -5917068, -433817613, ...];

%e n=4: [1, 8, 42, 160, 497, 0, -90536, -6434272, -496083426, ...];

%e n=5: [1, 15, 130, 810, 4075, 16929, 0, -5638950, -504633465, ...];

%e n=6: [1, 24, 315, 2920, 21396, 132264, 707500, 0, -412691760, ...];

%e n=7: [1, 35, 651, 8435, 85225, 716457, 5290089, 35515563, 0, ...]; ...

%e illustrating that the coefficient of x^(n+1) in A(x)^(n^2) / (x*A(x)^n)' equals 0 for n>=0.

%o (PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A = concat(A, 0); A[#A] = Vec( Ser(A)^((#A-2)^2)/(x*Ser(A)^(#A-2))' )[#A]/if(#A==2,1,2*(#A-2)) ); A[n+1]}

%o for(n=0, 20, print1(a(n), ", "))

%Y Cf. A300995.

%K nonn

%O 0,5

%A _Paul D. Hanna_, Apr 05 2018

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Last modified January 26 22:13 EST 2023. Contains 359836 sequences. (Running on oeis4.)