|
%I #16 Mar 11 2021 10:15:37
%S 1,480,2245320,43083161600,2331513459843750,287128730182879382976,
%T 69929145078323834449039740,30496052356323314014140611297280,
%U 22113924320024426907851753695581691875,25177421842925471123473548283955430812500000,42994775028354266041451477298870703788676694998956,106089234738948935762581435147478647028049918327743508480
%N O.g.f. A(x) satisfies: [x^n] exp( n^5 * x ) * (1 - x*A(x)) = 0 for n > 0.
%C 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^5*x) * (1 - x*A(x)) is equal to 0.
%F a(n) = (n+1)^(5*n+5)/(n+1)! - Sum_{k=1..n} (n+1)^(5*k)/k! * a(n-k) for n > 0 with a(0) = 1.
%F a(n) = A342202(5,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)^(5*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 O.g.f.: A(x) = 1 + 480*x + 2245320*x^2 + 43083161600*x^3 + 2331513459843750*x^4 + 287128730182879382976*x^5 + 69929145078323834449039740*x^6 + ...
%o (PARI) /* From formula: [x^n] exp( n^5*x ) * (1 - x*A(x)) = 0 */
%o {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*m^5 +x^2*O(x^m)) * (1 - x*Ser(A)) )[m+1] ); A[n+1]}
%o for(n=0, 20, print1( a(n), ", "))
%Y Cf. A304325, A304397, A107668, A107675, A304394.
%Y INVERT transform of A304325.
%K nonn
%O 0,2
%A _Paul D. Hanna_, May 12 2018
| 