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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)!. 2

%I #13 Jul 03 2014 04:19:50

%S 1,1,2,14,196,4652,166168,8232296,535974416,44186331248,4489336764064,

%T 550549455440096,80153857492836928,13665883723351362752,

%U 2697370187692768024448,610301579538939633684608,156933087218604923576672512,45515622704384079509089136384,14789652457653705738777659937280

%N 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 Vaclav Kotesovec, <a href="/A244577/b244577.txt">Table of n, a(n) for n = 0..160</a>

%F Given g.f. A(x), Sum_{k=0..n} [x^k] A(x)^n = (n+1)!.

%F a(n) ~ exp(-1) * (n!)^2. - _Vaclav Kotesovec_, Jul 03 2014

%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 14*x^3/3! + 196*x^4/4! + 4652*x^5/5! +...

%e ILLUSTRATION OF INITIAL TERMS.

%e If we form an array of coefficients of x^k/k! in A(x)^n, n>=0, like so:

%e A^0: [1],0, 0, 0, 0, 0, 0, 0, 0, ...;

%e A^1: [1, 1], 2, 14, 196, 4652, 166168, 8232296, 535974416, ...;

%e A^2: [1, 2, 6], 40, 528, 11824, 403840, 19373792, 1232259840, ...;

%e A^3: [1, 3, 12, 84], 1068, 22716, 741456, 34375200, 2132407248, ...;

%e A^4: [1, 4, 20, 152, 1912], 39008, 1218496, 54513152, 3292657664, ...;

%e A^5: [1, 5, 30, 250, 3180, 62980],1889080, 81499400, 4785873360, ...;

%e A^6: [1, 6, 42, 384, 5016, 97632, 2826288],117620256, 6706638336, ...;

%e A^7: [1, 7, 56, 560, 7588, 146804, 4127200, 165911312], 9177810320, ...;

%e A^8: [1, 8, 72, 784, 11088, 215296, 5918656, 230372480, 12358846848], ...; ...

%e 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 1! = 1;

%e 2! = 1 + 1;

%e 3! = 1 + 2 + 6/2!;

%e 4! = 1 + 3 + 12/2! + 84/3!;

%e 5! = 1 + 4 + 20/2! + 152/3! + 1912/4!;

%e 6! = 1 + 5 + 30/2! + 250/3! + 3180/4! + 62980/5!; ...

%o (PARI) /* By Definition (slow): */

%o {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 for(n=0, 20, print1(a(n), ", "))

%o (PARI) /* Faster, using series reversion: */

%o {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 for(i=1, n, G = 1 + intformal( (B-1)*G/x - B*G^2)); n!*polcoeff(x/serreverse(x*G), n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A244589, A232606, A232607, A232683, A232687.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jun 30 2014

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