A180749 - OEIS

%I #13 Jan 22 2018 11:00:02

%S 1,2,5,26,231,2844,43854,803578,16960731,404010692,10705681566,

%T 312189558548,9933838621998,342530711507568,12724338381577576,

%U 506700249728722586,21535304484380633171,973107015782753948460

%N G.f.: A(x) = x/Series_Reversion(x*G(x)) where G(x) = Sum_{n>=0} (n+1)^n*x^n.

%H Paul D. Hanna, <a href="/A180749/b180749.txt">Table of n, a(n) for n = 0..300</a>

%F G.f. satisfies: [x^n] A(x)^(n+1) = (n+1)^(n+1) for n>=0.

%F G.f. satisfies: A(x) = G(x/A(x)) where A(x*G(x)) = G(x) = Sum_{n>=0} (n+1)^n*x^n.

%e G.f.: A(x) = 1 + 2*x + 5*x^2 + 26*x^3 + 231*x^4 + 2844*x^5 +...

%e G.f. satisfies A(x) = G(x/A(x)) where A(x*G(x)) = G(x) begins:

%e G(x) = 1 + 2*x + 3^2*x^2 + 4^3*x^3 + 5^4*x^4 + 6^5*x^5 + 7^6*x^6 +...

%e so that:

%e A(x) = 1 + 2*x/A(x) + 3^2*x^2/A(x)^2 + 4^3*x^3/A(x)^3 + 5^4*x^4/A(x)^4 +...

%e The coefficients in A(x)^n for n=1..8 begin:

%e A^1: [(1), 2,   5,   26,  231,  2844,   43854,    803578, ...];

%e A^2: [1,  (4), 14,   72,  591,  6872,  102070,   1823024, ...];

%e A^3: [1,   6, (27), 146, 1140, 12546,  179105,   3112332, ...];

%e A^4: [1,   8,  44, (256),1954, 20488,  280848,   4740128, ...];

%e A^5: [1,  10,  65,  410,(3125),31512,  415020,   6793750, ...];

%e A^6: [1,  12,  90,  616, 4761,(46656), 591638,   9384288, ...];

%e A^7: [1,  14, 119,  882, 6986, 67214, (823543), 12652712, ...];

%e A^8: [1,  16, 152, 1216, 9940, 94768, 1126992, (16777216), ...]; ...

%e where the coefficient of x^n in A(x)^(n+1) equals (n+1)^(n+1).

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

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

%Y Cf. A182957, A180747, A242749.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 22 2011