A177752 a(n) = [x^n] A(x)^n/(n+1) for n>1 where g.f. A(x) = Sum_{n>=0} a(n)*x^n with A(0)=A'(0)=1.

1, 1, 1, 7, 103, 2461, 85236, 4017959, 247296043, 19273646845, 1856844859198, 216816668773915, 30183137125497500, 4940046471894678145, 939290181452023884094, 205341788464993702617408

Offset

0,4

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Formula

Given g.f. A(x), if D(x) satisfies:

. D(x) = A(x*D(x)), so that

. [x^n] D(x) = [x^n] A(x)^(n+1)/(n+1),

then in this case:

. D(x) = exp( Sum_{n>=1} (n+1)*a(n)*x^n/n - x).

a(n) ~ c * (n!)^2 / sqrt(n), where c = 0.500612869985729164508780668394780439... - Vaclav Kotesovec, Oct 18 2017

Example

G.f.: A(x) = 1 + x + x^2 + 7*x^3 + 103*x^4 + 2461*x^5 + 85236*x^6 +...
Coefficients in the initial powers of A(x) begin:
[1, 1, 1, 7, 103, 2461, 85236, 4017959, 247296043,...];
[1, 2,(3), 16, 221, 5142, 175649, 8212754, 502843539,...];
[1, 3, 6,(28), 357, 8067, 271591, 12592554, 766917744,...];
[1, 4, 10, 44,(515), 11264, 373448, 17166028, 1039805373,...];
[1, 5, 15, 65, 700,(14766), 481645, 21942390, 1321805265,...];
[1, 6, 21, 92, 918, 18612,(596652), 26931450, 1613229105,...];
[1, 7, 28, 126, 1176, 22848, 718991,(32143672), 1914402210,...];
[1, 8, 36, 168, 1482, 27528, 849244, 37590240,(2225664387),...]; ...
where the above coefficients in parenthesis illustrate the property
that the coefficient of x^n in A(x)^n = (n+1)*a(n) for n>1:
3 = 3*1; 28 = 4*7; 515 = 5*103; 14766 = 6*2461; 596652 = 7*85236; 32143672 = 8*4017959; ...
Therefore the logarithmic derivative of D(x) = A(x*D(x)) equals:
D'(x)/D(x) = 1 + 3*x + 28*x^2 + 515*x^3 + 14766*x^4 + 596652*x^5 +...
where [x^n] D(x) = [x^n] A(x)^(n+1)/(n+1) yields
D(x) = 1 + x + 2*x^2 + 11*x^3 + 140*x^4 + 3102*x^5 + 102713*x^6 +...

Mathematica

a[n_] := a[n] = SeriesCoefficient[Sum[a[k]*x^k, {k, 0, n-1}]^n, {x, 0, n}]; a[0] = 1; a[1] = 1; Table[a[n], {n, 0, 20}] (* Vaclav Kotesovec, Oct 18 2017 *)

Prog

(PARI) {a(n)=local(F=1+sum(m=1, n-1, a(m)*x^m)+x*O(x^n)); if(n<2, 1, polcoeff(F^n, n))}

Crossrefs

Cf. A177753, A293864.

Sequence in context: A365014 A234292 A357347 * A101746 A318398 A318815

Adjacent sequences: A177749 A177750 A177751 * A177753 A177754 A177755

Keyword

nonn

Author

Paul D. Hanna, May 13 2010

Status

approved