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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.
6
1, 112, 76221, 152978176, 673315202500, 5508710472669120, 75300988091046198131, 1595530380622638283804672, 49561200934127182294698009969, 2161539625780059763174286300310000, 127884966535158110582342524738392563401, 9979510403062963314615799917574094659938048, 1003426348756281631241586585232930123009989117616
OFFSET
0,2
COMMENTS
INVERT transform of A304324.
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.
FORMULA
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.
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
EXAMPLE
O.g.f.: A(x) = 1 + 112*x + 76221*x^2 + 152978176*x^3 + 673315202500*x^4 + 5508710472669120*x^5 + 75300988091046198131*x^6 + ...
PROG
(PARI) /* From formula: [x^n] exp( n^4*x ) * (1 - x*A(x)) = 0 */
{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]}
for(n=0, 25, print1( a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 12 2018
STATUS
approved