A375452 Expansion of g.f. A(x) satisfying [x^n] x*A'(x) / (1 - n*A(x)) = n^n for n >= 1.

1, 1, 3, 18, 170, 2181, 34909, 663152, 14493060, 356894730, 9757244200, 292895688618, 9570802060144, 338073560791701, 12835113747145545, 521191511664816256, 22541443155867228040, 1034573007453548334066, 50226021692317694266696, 2571751515973758412301940

Offset: 1

Author

Paul D. Hanna, Sep 09 2024

Keywords

• nonn

Comments

Conjecture: for n > 1, a(n) is odd iff n is of the form 2^k-2 or 2^k-1 for some k > 1.

Compare to [x^n] x*G'(x) / (1 - n*G(x)) = ((n+1)^n - 1)/n for n >= 1 when G(x) = x/(1-x).

Examples

			G.f.: A(x) = x + x^2 + 3*x^3 + 18*x^4 + 170*x^5 + 2181*x^6 + 34909*x^7 + 663152*x^8 + 14493060*x^9 + 356894730*x^10 + 9757244200*x^11 + 292895688618*x^12 + ...
where [x^n] x*A'(x) / (1 - n*A(x)) = n^n for n >= 1.
The table of coefficients of x^k in x*A'(x)/(1 - n*A(x)) begins
  n=1: [1, 3, 13,  91,  981, 14421, 262963, ...];
  n=2: [1, 4, 19, 124, 1196, 16324, 286553, ...];
  n=3: [1, 5, 27, 177, 1561, 19323, 319159, ...];
  n=4: [1, 6, 37, 256, 2166, 24330, 368887, ...];
  n=5: [1, 7, 49, 367, 3125, 32761, 450563, ...];
  n=6: [1, 8, 63, 516, 4576, 46656, 589093, ...];
  n=7: [1, 9, 79, 709, 6681, 68799, 823543, ...];
  ...
in which the main diagonal equals n^n for n >= 1.
RELATED SERIES.
x*A'(x)/(1 - A(x)) = x + 3*x^2 + 13*x^3 + 91*x^4 + 981*x^5 + 14421*x^6 + 262963*x^7 + 5630843*x^8 + ...
Let B(x) satisfy A(x/B(x)) = x, then B(x) begins
B(x) = 1 + x + 2*x^2 + 11*x^3 + 105*x^4 + 1375*x^5 + 22390*x^6 + 430954*x^7 + 9512029*x^8 + ... + A375457(n)*x^n + ...
Below we demonstrate some properties of this related series.
The table of coefficients in B(x)^n begins:
  n=1: [1, 1,  2,  11,  105,  1375, ...];
  n=2: [1, 2,  5,  26,  236,  3004, ...];
  n=3: [1, 3,  9,  46,  399,  4932, ...];
  n=4: [1, 4, 14,  72,  601,  7212, ...];
  n=5: [1, 5, 20, 105,  850,  9906, ...];
  ...
in which the sum of the first n coefficients in B(x)^n equals the coefficients in x*A'(x)/(1 - A(x)) like so: 1 = 1, 3 = 1 + 2, 13 = 1 + 3 + 9, 91 = 1 + 4 + 14 + 72, 981 = 1 + 5 + 20 + 105 + 850, etc.
Also, the sum of the first n coefficients in B(x/n)^n equals n, as illustrated by
 1 = 1;
 2 = 1 + 2/2;
 3 = 1 + 3/3 + 9/3^2;
 4 = 1 + 4/4 + 14/4^2 + 72/4^3;
 5 = 1 + 5/5 + 20/5^2 + 105/5^3 + 850/5^4;
 ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..300

Crossrefs

Cf. A375449, A375457.

Programs

PARI

{a(n) = my(A=[0,1],Ax=x,m); for(i=1,n, A = concat(A,0); Ax = Ser(A); m=#A-1; A[#A] = (m^m - polcoef( x*Ax' / (1 - m*Ax), m) )/m ); A[n+1]}
for(n=1,20,print1(a(n),", "))

Formula

a(n) ~ (1 - exp(-1)) * n^(n-1). - Vaclav Kotesovec, Sep 10 2024