A305114 O.g.f. A(x) satisfies: [x^n] exp( x*A(x) ) * (n^2 + 1 - A(x)) = 0 for n >= 0.

1, 1, 5, 49, 818, 20902, 761661, 37594453, 2417371010, 196435255206, 19688467059690, 2386332383219634, 344073696796404628, 58206664284882779756, 11417574208958284857661, 2570602844156601626700525, 658420262904330345422925698, 190366426847612777404133776150, 61701278611229563703556400453662, 22281060851931816259923863246915334, 8914783049343376043121712182459906492

Offset: 0

Author

Paul D. Hanna, May 26 2018

Keywords

• nonn

Comments

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

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

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

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

Examples

			O.g.f.: A(x) = 1 + x + 5*x^2 + 49*x^3 + 818*x^4 + 20902*x^5 + 761661*x^6 + 37594453*x^7 + 2417371010*x^8 + 196435255206*x^9 + 19688467059690*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( x*A(x) ) * (n^2 + 1 - A(x)) begins:
n=0: [0, -1, -12, -333, -21136, -2625525, -565374636, ...];
n=1: [1, 0, -9, -296, -19815, -2520504, -549669785, ...];
n=2: [4, 3, 0, -185, -15852, -2205441, -502555232, ...];
n=3: [9, 8, 15, 0, -9247, -1680336, -424030977, ...];
n=4: [16, 15, 36, 259, 0, -945189, -314097020, ...];
n=5: [25, 24, 63, 592, 11889, 0, -172753361, ...];
n=6: [36, 35, 96, 999, 26420, 1155231, 0, ...];
n=7: [49, 48, 135, 1480, 43593, 2520504, 204163063, 0, ...]; ...
in which the coefficients of x^n in row n form a diagonal of zeros.
The secondary diagonal [1, 3, 15, 259, 11889, 1155231, 204163063, ...] equals [1, 3*1, 5*3, 7*37, 9*1321, 11*105021, 13*15704851, 15*3951915073, ...], which equals the odd numbers multiplied by the coefficients in exp(x*A(x)) as shown below.
RELATED SERIES.
exp(x*A(x)) = 1 + x + 3*x^2/2! + 37*x^3/3! + 1321*x^4/4! + 105021*x^5/5! + 15704851*x^6/6! + 3951915073*x^7/7! + 1548047295537*x^8/8! + ... + E(n)*x^n/n! + ... where E(n) = (1/n^2) * Sum_{k=1..n} n!/(n-k)! * a(k) * E(n-k) for n > 0.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A305110, A305112, A305113, A305115, A305116.

Programs

PARI

{a(n) = my(A=[1],m); for(i=1,n, A=concat(A,0); m=#A; A[m] = Vec( exp( x*(Ser(A)) ) * ((m-1)^2+1 - Ser(A)) )[m] ); A[n+1]}
for(n=0,20,print1(a(n),", "))
/* Routine to generate E(n) where exp(x*A(x)) = Sum_{n>=0} E(n) * x^n/n! */
{E(n) = if(n==0,1, (1/n^2) * sum(k=1,n, n!/(n-k)! * a(k) * E(n-k) ))}
for(n=0,20,print1(E(n),", "))

Formula

E(n) = (1/n^2) * Sum_{k=1..n} n!/(n-k)! * a(k) * E(n-k) for n > 0 with E(0) = 1, where exp(x*A(x)) = Sum_{n>=0} E(n) * x^n/n!.

a(n) ~ c * n!^2, where c = 1.509843562791802670961526317096279216134... - Vaclav Kotesovec, Oct 06 2020