A304396 - OEIS

%I #12 Mar 24 2026 17:42:15

%S 1,7,941,597571,1077304324,4250548204220,31362344061243731,

%T 389533784331698799757,7553909607396308839307729,

%U 216153962578005976317428630031,8734715288242477329577387114158361,481264969283514820342197141086713669943,35132745658635258962977017094392007388046256,3317919567828983194629673950155604531470409170896

%N a(n) = A304394(n) / (n+1)^4.

%F a(n) = (n+1)^(4*n)/(n+1)! - Sum_{k=1..n} (n+1)^(4*k-4)/k! * (n-k+1)^4 * a(n-k) for n>0 with a(0)=1.

%F From _Seiichi Manyama_, Mar 24 2026: (Start)

%F 1/(n+1)! = Sum_{k=0..n} ((k+1)/(n+1)^(k+1))^4 * a(k)/(n-k)!.

%F G.f. A(x) satisfies [x^n] exp(n^4*x) (1 - E^4(x*A(x))) = 0 for n > 0, where E is the Euler operator x*d/dx. (End)

%o (PARI) /* A304394 formula: [x^n] exp( n^4*x ) * (1 - x*A(x)) = 0 */

%o {A304394(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 for(n=0, 25, print1( A304394(n)/(n+1)^4, ", "))

%Y Cf. A107669, A304394, A304397.

%K nonn

%O 0,2

%A _Paul D. Hanna_, May 12 2018