A304859 O.g.f. A(x) satisfies: [x^n] exp( x*A(x) ) * ((n+1)^3 - A(x)) = 0 for n >= 0.

1, 7, 188, 12055, 1506778, 325137986, 111398218424, 56980772011046, 41506384038800428, 41480331680194025916, 55182540633987204679584, 95316642195992777390509848, 209341095877177204596122425408, 574274492811735528719934388002880, 1937733557293593354972486212461808912, 7935430935079838085823331455696837604360, 38980412505942014324178479481443585982598736

Offset

0,2

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.

Links

Table of n, a(n) for n=0..16.

Formula

E(n) = (1/((n+1)^3-1)) * 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!.

Example

O.g.f.: A(x) = 1 + 7*x + 188*x^2 + 12055*x^3 + 1506778*x^4 + 325137986*x^5 + 111398218424*x^6 + 56980772011046*x^7 + 41506384038800428*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( x*A(x) ) * ((n+1)^3 - A(x)) begins:
n=0: [0, -7, -390, -73773, -36518620, -39222931815, -80459970347322, ...];
n=1: [7, 0, -285, -65576, -34457085, -37945906128, -78813128848985, ...];
n=2: [26, 19, 0, -43327, -28861490, -34479693549, -74343130496356, ...];
n=3: [63, 56, 555, 0, -17964805, -27729700632, -65638396862289, ...];
n=4: [124, 117, 1470, 71431, 0, -16601333931, -51287349519638, ...];
n=5: [215, 208, 2835, 177992, 26799955, 0, -29878410041257, ...];
n=6: [342, 335, 4740, 326709, 64202090, 23168894607, 0, ...];
n=7: [511, 504, 7275, 524608, 113973435, 53999943336, 39759459031279, 0, ...]; ...
in which the main diagonal is all zeros, illustrating the exponential property that [x^n] exp( x*A(x) ) * ((n+1)^3 - A(x)) = 0 for n >= 0.
RELATED SERIES.
exp(x*A(x)) = 1 + x + 15*x^2/2! + 1171*x^3/3! + 294505*x^4/4! + 182432241*x^5/5! + 235263071191*x^6/6! + 563154646240915*x^7/7! + 2302069522660831281*x^8/8! + ... + E(n)*x^n/n! + ... where E(n) = (1/((n+1)^3-1)) * Sum_{k=1..n} n!/(n-k)! * a(k) * E(n-k) for n > 0.

Prog

(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^3 - 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+1)^3-1)) * sum(k=1, n, n!/(n-k)! * a(k) * E(n-k) ))}
for(n=0, 20, print1(E(n), ", "))

Crossrefs

Cf. A304858, A305110, A305112, A305113, A305114, A305115, A305116.

Sequence in context: A296234 A219567 A202791 * A303292 A010332 A368592

Adjacent sequences: A304856 A304857 A304858 * A304860 A304861 A304862

Keyword

nonn

Author

Paul D. Hanna, May 26 2018

Status

approved