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A304394 O.g.f. A(x) satisfies: [x^n] exp(n^4 * x) * (1 - x*A(x)) = 0 for n > 0.

%I A304394 #22 Mar 11 2021 17:43:25
%S A304394 1,112,76221,152978176,673315202500,5508710472669120,
%T A304394 75300988091046198131,1595530380622638283804672,
%U A304394 49561200934127182294698009969,2161539625780059763174286300310000,127884966535158110582342524738392563401,9979510403062963314615799917574094659938048,1003426348756281631241586585232930123009989117616
%N A304394 O.g.f. A(x) satisfies: [x^n] exp(n^4 * x) * (1 - x*A(x)) = 0 for n > 0.
%C A304394 INVERT transform of A304324.
%C A304394 The o.g.f. A(x) = Sum_{m >= 0} a(m)*x^m is such that, for each integer n > 0, the coefficient of x^n in the expansion of exp(n^4 * x) * (1 - x*A(x)) = 0 is equal to 0.
%F A304394 a(n) = (n+1)^(4*n+4)/(n+1)! - Sum_{k=1..n} (n+1)^(4*k)/k! * a(n-k) for n > 0 with a(0) = 1.
%F A304394 a(n) = A342202(4,n+1) = Sum_{r=1..(n+1)} (-1)^(r-1) * Sum_{s_1, ..., s_r} (1/(Product_{j=1..r} s_j!)) * Product_{j=1..r} (Sum_{i=1..j} s_i)^(4*s_j)), where the second sum is over lists (s_1, ..., s_r) of positive integers s_i such that Sum_{i=1..r} s_i = n+1. (Thus the second sum is over all compositions of n+1. See _Michel Marcus_'s PARI program in A342202.) - _Petros Hadjicostas_, Mar 10 2021
%e A304394 O.g.f.: A(x) = 1 + 112*x + 76221*x^2 + 152978176*x^3 + 673315202500*x^4 + 5508710472669120*x^5 + 75300988091046198131*x^6 + ...
%o A304394 (PARI) /* From formula: [x^n] exp( n^4*x ) * (1 - x*A(x)) = 0 */
%o A304394 {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*m^4 +x^2*O(x^m)) * (1 - x*Ser(A)) )[m+1] ); A[n+1]}
%o A304394 for(n=0, 25, print1( a(n), ", "))
%Y A304394 Cf. A304324, A304396, A107668, A107675, A304395, A342202.
%K A304394 nonn
%O A304394 0,2
%A A304394 _Paul D. Hanna_, May 12 2018