A219270 - OEIS

%I #7 Nov 18 2012 20:24:52

%S 1,1,3,19,357,36301,25099919,125586786039,5057589140280713,

%T 1834978988294513263993,6658624933768189847669906875,

%U 265790340541205664455435665865706651,127313966488883223582008076686428073754622381,792786699250878346141762323574616271109902136066501317

%N G.f. satisfies: A(x) = Sum_{n>=0} x^n * A(x)^n * Product_{k=0..n} k!.

%F a(n) = [x^n] F(x)^(n+1)/(n+1) where F(x) = Sum_{n>=0} x^n*Product_{k=0..n} k! is the g.f. of the superfactorials A000178.

%F G.f.: (1/x)*Series_Reversion(x/F(x)) where F(x) = Sum_{n>=0} x^n*Product_{k=0..n} k! is the g.f. of A000178.

%e G.f.: A(x) = 1 + x + 3*x^2 + 19*x^3 + 357*x^4 + 36301*x^5 + 25099919*x^6 +...

%e where

%e A(x) = 1 + 1!*x*A(x) + 2!*1!*A(x)^2 + 3!*2!*1!*A(x)^3 + 4!*3!*2!*1!*A(x)^4 +...

%e Given F(x) is the g.f. of the superfactorials A000178:

%e F(x) = 1 + x + 2*x^2 + 12*x^3 + 288*x^4 + 34560*x^5 + 24883200*x^6 +...

%e then the table of coefficients in F(x)^n begins:

%e n=1: [(1), 1, 2, 12, 288, 34560, 24883200, 125411328000, ...];

%e n=2: [1,(2), 5, 28, 604, 69744, 49836816, 250872567552, ...];

%e n=3: [1, 3, (9), 49, 954, 105600, 74861864, 376383826368, ...];

%e n=4: [1, 4, 14, (76), 1345, 142184, 99959432, 501945213632, ...];

%e n=5: [1, 5, 20, 110,(1785), 179561, 125130690, 627556840100, ...];

%e n=6: [1, 6, 27, 152, 2283,(217806), 150376901, 753218818212, ...];

%e n=7: [1, 7, 35, 203, 2849, 257005,(175699433), 878931262217, ...];

%e n=8: [1, 8, 44, 264, 3494, 297256, 201099772,(1004694288312), ...]; ...

%e in which the main diagonal generates this sequence:

%e [1/1, 2/2, 9/3, 76/4, 1785/5, 217806/6, 175699433/7, 1004694288312/8, ...].

%o (PARI) {a(n)=polcoeff(1/x*serreverse(x/sum(k=0,n+1,prod(j=0,k,j!)*x^k)+x^2*O(x^n)),n)}

%o for(n=0,21,print1(a(n),", "))

%Y Cf. A219269, A000178.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 16 2012