A242749 G.f. satisfies: A(x) = G(x/A(x)) such that A(x*G(x)) = G(x) = Sum_{n>=0} (n+1)^(n+1)*x^n.

1, 4, 11, 60, 611, 8632, 151538, 3132140, 73883667, 1949844168, 56785116742, 1806695366616, 62314198956510, 2315470815127792, 92214156916779444, 3918743752606940812, 177018691811732542595, 8471087431826716955880, 428141645771934036086942, 22791557465710675500959688

Offset

0,2

Links

Table of n, a(n) for n=0..19.

Formula

G.f. satisfies: [x^n] A(x)^(n+1) = (n+1)^(n+2).

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

Example

G.f.: A(x) = 1 + 4*x + 11*x^2 + 60*x^3 + 611*x^4 + 8632*x^5 + 151538*x^6 +...
such that A(x*G(x)) = G(x) where:
G(x) = 1 + 4*x + 27*x^2 + 256*x^3 + 3125*x^4 +...+ (n+1)^(n+1)*x^n +...
also, A(x) = G(x/A(x)):
A(x) = 1 + 4*x/A(x) + 27*x^2/A(x)^2 + 256*x^3/A(x)^3 + 3125*x^4/A(x)^4 +...+ (n+1)^(n+1)*x^n/A(x)^n +...
If we form a table of coefficients of x^k in A(x)^n like so:
[1,  4,  11,   60,   611,   8632,  151538,   3132140, ...];
[1,  8,  38,  208,  1823,  23472,  389174,   7739808, ...];
[1, 12,  81,  508,  4164,  48852,  759407,  14463624, ...];
[1, 16, 140, 1024,  8418,  91920, 1335712,  24248640, ...];
[1, 20, 215, 1820, 15625, 163664, 2232620,  38498580, ...];
[1, 24, 306, 2960, 27081, 279936, 3623894,  59297664, ...];
[1, 28, 413, 4508, 44338, 462476, 5764801,  89716400, ...];
[1, 32, 536, 6528, 69204, 739936, 9018480, 134217728, ...]; ...
then the main diagonal forms the sequence A007778:
[1, 8, 81, 1024, 15625, 279936, 5764801, 134217728, ..., (n+1)^(n+2), ...].

Prog

(PARI) {a(n)=polcoeff(x/serreverse(x*sum(m=0, n+1, (m+1)^(m+1)*x^m)+x^2*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A180749, A007778.

Sequence in context: A245545 A002831 A246598 * A303955 A114053 A266386

Adjacent sequences: A242746 A242747 A242748 * A242750 A242751 A242752

Keyword

nonn

Author

Paul D. Hanna, May 21 2014

Status

approved