A303063 G.f. A(x) satisfies: [x^(n-1)] (1 + x*A(x)^n)^n / A(x)^n = 0 for n>1.

1, 1, 3, 17, 151, 1812, 26766, 461302, 8978490, 193200156, 4529641423, 114510000515, 3097375627215, 89116723381943, 2714808312021989, 87242980758842543, 2948618278635037930, 104544558380661516685, 3880035778583841094470, 150451784852703095162304, 6084892588256393044757197, 256294338370540915598727500

Offset: 0

Author

Paul D. Hanna, Apr 17 2018

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 151*x^4 + 1812*x^5 + 26766*x^6 + 461302*x^7 + 8978490*x^8 + 193200156*x^9 + 4529641423*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients in (1 + x*A(x)^n)^n / A(x)^n begins:
n=1: [1, 0, -2, -12, -116, -1475, -22625, -400078, ...];
n=2: [1, 0, -2, -18, -197, -2630, -41347, -742194, ...];
n=3: [1, 0, 0, -15, -228, -3390, -55716, -1022901, ...];
n=4: [1, 0, 4, 0, -178, -3536, -64144, -1228756, ...];
n=5: [1, 0, 10, 30, 0, -2640, -63025, -1327450, ...];
n=6: [1, 0, 18, 78, 369, 0, -45519, -1252758, ...];
n=7: [1, 0, 28, 147, 1008, 5425, 0, -881412, ...];
n=8: [1, 0, 40, 240, 2012, 15080, 91832, 0, ...]; ...
in which the main diagonal equals all zeros after the initial term, illustrating that [x^(n-1)] (1 + x*A(x)^n)^n / A(x)^n = 0 for n>1.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A303062, A360578, A360581.

Programs

PARI

{a(n) = my(A=[1]); for(m=1,n+1, A=concat(A,0); A[m] = Vec( (1 + x*Ser(A)^m)^m/Ser(A)^m )[m]/m ); A[n+1]}
for(n=0,30, print1(a(n),", "))

Formula

a(n) ~ c * n! * n^(3*LambertW(1) + 1/(1 + LambertW(1))) / LambertW(1)^n, where c = 0.03203091421745281863810572012... - Vaclav Kotesovec, Aug 11 2021