A392213 G.f. A(x) satisfies [x^n] A(x)^prime(n) = prime(n) * n^n for n >= 1.

1, 1, 3, 13, 101, 333, 18155, 141203, 6544864, 84442385, 1310288057, 114250367488, 1002914188652, 89834152777920, 3336675861288631, 99030385161776357, 3757185586490106605, 178830667844799920927, 11941685503176695567135, 310955524618383919196635, 28448126980292502958222854, 1519932779827609052168632180

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..500

Example

G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 101*x^4 + 333*x^5 + 18155*x^6 + 141203*x^7 + 6544864*x^8 + 84442385*x^9 + 1310288057*x^10 + ...
Below we illustrate the property [x^n] A(x)^prime(n) = prime(n)*n^n for n >= 0; here prime(0) is taken to be 1.
The table of coefficients of x^k in A(x)^prime(n) begins
  n = 0: [1,  1,   3,   13,   101,    333,   18155, ...];
  n = 1: [1,  2,   7,   32,   237,    946,   37751, ...];
  n = 2: [1,  3,  12,   58,   417,   1905,   59352, ...];
  n = 3: [1,  5,  25,  135,   950,   5186,  111370, ...];
  n = 4: [1,  7,  42,  252,  1792,  10962,  181118, ...];
  n = 5: [1, 11,  88,  638,  4851,  34375,  414392, ...];
  n = 6: [1, 13, 117,  923,  7332,  54912,  606528, ...]; ...
in which the main diagonal equals A180353 (with offset):
[1, 2, 12, 135, 1792, 34375, 606528, ..., prime(n)*n^n, ...].

Prog

(PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A = concat(A, 0); m = #A-1;
A[#A] = m^m - polcoef(Ser(A)^prime(m), m)/prime(m) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A392214, A182957, A381353, A180353 (prime(n)*n^n).

Sequence in context: A267196 A268215 A323687 * A338697 A168417 A352170

Adjacent sequences: A392210 A392211 A392212 * A392214 A392215 A392216

Keyword

nonn

Author

Paul D. Hanna, Feb 03 2026

Status

approved