A305113 O.g.f. A(x) satisfies: [x^n] exp( n * x*A(x) ) * (n + 3 - A(x)) = 0 for n > 0.

1, 3, 26, 429, 11140, 413575, 20732442, 1349324599, 110687183288, 11178507440925, 1363390073347730, 197621842687876673, 33583706596253821788, 6613771286589113270611, 1494257702398724584231946, 383919965020249799260370355, 111309986933239661174592643760, 36168040052169465186266712298305, 13090361918944129697134824216839154, 5248357548475200108500930014786066693, 2319429294524000405318027925169538777460

Offset

0,2

Comments

Note: the factorial series, F(x) = Sum_{n>=0} n! * x^n, satisfies:

(1) [x^n] exp( n * x*F(x) ) * (2 - F(x)) = 0 for n > 0,

(2) [x^n] exp( x*F(x) ) * (n + 1 - F(x)) = 0 for n > 0.

It is remarkable that this sequence should consist entirely of integers.

Links

Table of n, a(n) for n=0..20.

Example

O.g.f.: A(x) = 1 + 3*x + 26*x^2 + 429*x^3 + 11140*x^4 + 413575*x^5 + 20732442*x^6 + 1349324599*x^7 + 110687183288*x^8 + 11178507440925*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n * x*A(x) ) * (n + 3 - A(x)) begins:
n=0: [2, -3, -52, -2574, -267360, -49629000, -14927358240, ...];
n=1: [3, 0, -37, -2268, -248745, -47203552, -14374490745, ...];
n=2: [4, 5, 0, -1462, -202352, -41536632, -13142258240, ...];
n=3: [5, 12, 65, 0, -121911, -32155140, -11164894659, ...];
n=4: [6, 21, 164, 2298, 0, -18516616, -8369990496, ...];
n=5: [7, 32, 303, 5636, 172075, 0, -4677722165, ...];
n=6: [8, 45, 488, 10242, 404400, 24104328, 0, ...];
n=7: [9, 60, 725, 16368, 708573, 54605228, 5760470145, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating the property that [x^n] exp( n * x*A(x) ) * (n + 3 - A(x)) = 0 for n > 0.
RELATED SERIES.
exp(x*A(x)) = 1 + x + 7*x^2/2! + 175*x^3/3! + 11065*x^4/4! + 1399801*x^5/5! + 307183471*x^6/6! + 106838020087*x^7/7! + 55316481920785*x^8/8! + ...

Prog

(PARI) {a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)*x*(Ser(A)) ) * ((m-1) + 2 - Ser(A)) )[m] ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A305110, A305111, A305112, A305114.

Sequence in context: A206402 A300954 A122949 * A251664 A182958 A174423

Adjacent sequences: A305110 A305111 A305112 * A305114 A305115 A305116

Keyword

nonn

Author

Paul D. Hanna, May 26 2018

Status

approved