A304395 O.g.f. A(x) satisfies: [x^n] exp( n^5 * x ) * (1 - x*A(x)) = 0 for n > 0.

1, 480, 2245320, 43083161600, 2331513459843750, 287128730182879382976, 69929145078323834449039740, 30496052356323314014140611297280, 22113924320024426907851753695581691875, 25177421842925471123473548283955430812500000, 42994775028354266041451477298870703788676694998956, 106089234738948935762581435147478647028049918327743508480

Offset

0,2

Comments

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.

Links

Table of n, a(n) for n=0..11.

Formula

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.

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

Example

O.g.f.: A(x) = 1 + 480*x + 2245320*x^2 + 43083161600*x^3 + 2331513459843750*x^4 + 287128730182879382976*x^5 + 69929145078323834449039740*x^6 + ...

Prog

(PARI) /* From formula: [x^n] exp( n^5*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^5 +x^2*O(x^m)) * (1 - x*Ser(A)) )[m+1] ); A[n+1]}
for(n=0, 20, print1( a(n), ", "))

Crossrefs

Cf. A304325, A304397, A107668, A107675, A304394.

INVERT transform of A304325.

Sequence in context: A289744 A287933 A362215 * A269759 A317174 A020286

Adjacent sequences: A304392 A304393 A304394 * A304396 A304397 A304398

Keyword

nonn

Author

Paul D. Hanna, May 12 2018

Status

approved