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A244577 G.f. A(x) satisfies the property that the sum of the coefficients of x^k, k=0..n, in A(x)^n equals (n+1)!.

%I A244577 #13 Jul 03 2014 04:19:50
%S A244577 1,1,2,14,196,4652,166168,8232296,535974416,44186331248,4489336764064,
%T A244577 550549455440096,80153857492836928,13665883723351362752,
%U A244577 2697370187692768024448,610301579538939633684608,156933087218604923576672512,45515622704384079509089136384,14789652457653705738777659937280
%N A244577 G.f. A(x) satisfies the property that the sum of the coefficients of x^k, k=0..n, in A(x)^n equals (n+1)!.
%H A244577 Vaclav Kotesovec, <a href="/A244577/b244577.txt">Table of n, a(n) for n = 0..160</a>
%F A244577 Given g.f. A(x), Sum_{k=0..n} [x^k] A(x)^n = (n+1)!.
%F A244577 a(n) ~ exp(-1) * (n!)^2. - _Vaclav Kotesovec_, Jul 03 2014
%e A244577 E.g.f.: A(x) = 1 + x + 2*x^2/2! + 14*x^3/3! + 196*x^4/4! + 4652*x^5/5! +...
%e A244577 ILLUSTRATION OF INITIAL TERMS.
%e A244577 If we form an array of coefficients of x^k/k! in A(x)^n, n>=0, like so:
%e A244577 A^0: [1],0,  0,   0,     0,      0,       0,         0,           0, ...;
%e A244577 A^1: [1, 1], 2,  14,   196,   4652,  166168,   8232296,   535974416, ...;
%e A244577 A^2: [1, 2,  6], 40,   528,  11824,  403840,  19373792,  1232259840, ...;
%e A244577 A^3: [1, 3, 12,  84], 1068,  22716,  741456,  34375200,  2132407248, ...;
%e A244577 A^4: [1, 4, 20, 152,  1912], 39008, 1218496,  54513152,  3292657664, ...;
%e A244577 A^5: [1, 5, 30, 250,  3180,  62980],1889080,  81499400,  4785873360, ...;
%e A244577 A^6: [1, 6, 42, 384,  5016,  97632, 2826288],117620256,  6706638336, ...;
%e A244577 A^7: [1, 7, 56, 560,  7588, 146804, 4127200, 165911312], 9177810320, ...;
%e A244577 A^8: [1, 8, 72, 784, 11088, 215296, 5918656, 230372480, 12358846848], ...; ...
%e A244577 then we can illustrate how the sum of the coefficients of x^k, k=0..n, in A(x)^n (shown above in brackets) equals (n+1)!:
%e A244577 1! = 1;
%e A244577 2! = 1 + 1;
%e A244577 3! = 1 + 2 + 6/2!;
%e A244577 4! = 1 + 3 + 12/2! + 84/3!;
%e A244577 5! = 1 + 4 + 20/2! + 152/3! + 1912/4!;
%e A244577 6! = 1 + 5 + 30/2! + 250/3! + 3180/4! + 62980/5!; ...
%o A244577 (PARI) /* By Definition (slow): */
%o A244577 {a(n)=if(n==0, 1, n!*((n+1)! - sum(k=0, n, polcoeff(sum(j=0, min(k, n-1), a(j)*x^j/j!)^n + x*O(x^k), k)))/n)}
%o A244577 for(n=0, 20, print1(a(n), ", "))
%o A244577 (PARI) /* Faster, using series reversion: */
%o A244577 {a(n)=local(B=sum(k=0, n+1, (k+1)!*x^k)+x^3*O(x^n), G=1+x*O(x^n));
%o A244577 for(i=1, n, G = 1 + intformal( (B-1)*G/x - B*G^2)); n!*polcoeff(x/serreverse(x*G), n)}
%o A244577 for(n=0, 30, print1(a(n), ", "))
%Y A244577 Cf. A244589, A232606, A232607, A232683, A232687.
%K A244577 nonn
%O A244577 0,3
%A A244577 _Paul D. Hanna_, Jun 30 2014