|
%I #9 Jun 06 2018 21:33:15
%S 1,7,188,12055,1506778,325137986,111398218424,56980772011046,
%T 41506384038800428,41480331680194025916,55182540633987204679584,
%U 95316642195992777390509848,209341095877177204596122425408,574274492811735528719934388002880,1937733557293593354972486212461808912,7935430935079838085823331455696837604360,38980412505942014324178479481443585982598736
%N O.g.f. A(x) satisfies: [x^n] exp( x*A(x) ) * ((n+1)^3 - A(x)) = 0 for n >= 0.
%C Note: the factorial series, F(x) = Sum_{n>=0} n! * x^n, satisfies:
%C (1) [x^n] exp( x*F(x) ) * (n + 1 - F(x)) = 0 for n > 0,
%C (2) [x^n] exp( n * x*F(x) ) * (2 - F(x)) = 0 for n > 0.
%C It is remarkable that this sequence should consist entirely of integers.
%F 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!.
%e 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 + ...
%e ILLUSTRATION OF DEFINITION.
%e The table of coefficients of x^k/k! in exp( x*A(x) ) * ((n+1)^3 - A(x)) begins:
%e n=0: [0, -7, -390, -73773, -36518620, -39222931815, -80459970347322, ...];
%e n=1: [7, 0, -285, -65576, -34457085, -37945906128, -78813128848985, ...];
%e n=2: [26, 19, 0, -43327, -28861490, -34479693549, -74343130496356, ...];
%e n=3: [63, 56, 555, 0, -17964805, -27729700632, -65638396862289, ...];
%e n=4: [124, 117, 1470, 71431, 0, -16601333931, -51287349519638, ...];
%e n=5: [215, 208, 2835, 177992, 26799955, 0, -29878410041257, ...];
%e n=6: [342, 335, 4740, 326709, 64202090, 23168894607, 0, ...];
%e n=7: [511, 504, 7275, 524608, 113973435, 53999943336, 39759459031279, 0, ...]; ...
%e 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.
%e RELATED SERIES.
%e 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.
%o (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]}
%o for(n=0,20,print1(a(n),", "))
%o /* Routine to generate E(n) where exp(x*A(x)) = Sum_{n>=0} E(n) * x^n/n! */
%o {E(n) = if(n==0, 1, (1/((n+1)^3-1)) * sum(k=1, n, n!/(n-k)! * a(k) * E(n-k) ))}
%o for(n=0, 20, print1(E(n), ", "))
%Y Cf. A304858, A305110, A305112, A305113, A305114, A305115, A305116.
%K nonn
%O 0,2
%A _Paul D. Hanna_, May 26 2018
|
